Lebesgue measure of distance sets with regular pins and multi-scale Mizohata-Takeuchi-type estimates

LEBESGUE MEASURE OF DIST ANCE SETS WITH REGULAR PINS AND MUL TI-SCALE MIZOHA T A-T AKEUCHI-TYPE ESTIMA TES BOCHEN LIU T o Pr ofessor Silei Wang my te acher of Calculus I and R e al A nalysis on the o c c asion of his 93r d birthday Abstract. Supp ose E , F are Borel sets in the plane, dim H E > 1, dim H E +dim H F > 2, and F has equal Hausdorff and pac king dimension. W e pro v e that there exists y ∈ F suc h that the pinned distance set ∆ y ( E ) := {| x − y | : x ∈ E } has p ositive Lebesgue measure. In particular, it settles the regular case of the distance set problem in the plane. The main ingredients of the proof consist of a multi-scale Go o d-Bad decomp osition and a multi-scale Mizohata-T akeuc hi-t ype estimate with ar- bitrary small p o w er-loss. Contents 1. In tro duction 1 2. Preliminaries 6 3. A multi-scale Goo d-Bad decomp osition 7 4. Multi-scale Mizohata-T ak euc hi-t yp e estimates 13 5. Pro of of the main Theorem 20 References 20 1. Intr oduction 1.1. F alconer distance conjecture. F alconer distance conjecture [16] is one of the most famous op en problems in harmonic analysis and geometric measure theory since 1985. It states that, if a Borel set E ⊂ R d , d ⩾ 2, has Hausdorff dimension dim H E > d/ 2, then its distance set ∆( E ) := {| x − y | : x, y ∈ E } has p ositiv e Leb esgue measure (denoted b y | ∆( E ) | > 0). A stronger v ersion is the pinned distance set problem, first studied by Peres and Schlag [38] in 2000, that asks whether there exists y ∈ E suc h that the pinned distance set ∆ y ( E ) := {| x − y | : x ∈ E } has p ositive Lebesgue measure when dim H E > d/ 2. This work is supp orted by the National Key R&D Program of China 2024YF A1015400, and the National Natural Science F oundation of China gran t 12131011. 1 2 BOCHEN LIU With more than 40 y ears passed, accum ulating efforts of man y excellen t mathemati- cians, F alconer distance conjecture is still op en in ev ery dimension. So far, there are t w o p ersp ectiv es to study this problem. One is to in v estigate ho w large dim H E needs to b e to ensure ∆( E ) or ∆ y ( E ) has p ositiv e Leb esgue measure. F alconer’s first result [16] in 1985 is of this t yp e, which is a L ∞ -estimate in to day’s p oint of view. In 1987, Mattila [32] proposed his L 2 -approac h for | ∆( E ) | , no w called the Mattila in tegral, that reduced the problem to spherical av eraging op erators. Since then, Bourgain [2], W olff [46], Erdogan [15], Du-Guth-Ou-W ang-Wilson-Zhang [10], Du-Zhang [14] hav e used the newly developed to ols in harmonic analysis at the time to impro v e spherical av eraging estimates for partial results on F alconer distance conjecture. In 2019, the author [29] proposed an L 2 -approac h for the pinned distance set problem. Shortly after, Guth, Iosevic h, Ou and W ang [25] prov ed the current b est-kno wn result on F alconer distance conjecture in the plane: dim H E > 5 / 4 = ⇒ | ∆ y ( E ) | > 0 , for some y ∈ E , building on the author’s approac h and an insigh tful use of the decoupling theory . Their argumen t in fact shows dim H F > 1 , 3 dim H E + dim H F > 5 = ⇒ | ∆ y ( E ) | > 0 , for some y ∈ F . W e shall review their pro of strategy in Section 3.1. This result w as later extended to all ev en dimensions by Du-Iosevich-Ou-W ang-Zhang [11]. F ollowing the same idea with more careful analysis, Du-Ou-Ren-Zhang [12][13] obtained the curren t best-known results in dimension 3 and higher. The other p ersp ective on F alconer distance conjecture is to compute the Hausdorff dimension of ∆( E ) , ∆ y ( E ) assuming dim H E > d/ 2. The first nontrivial result on ∆( E ) is due to Katz-T ao [27] and Bourgain [3] from discretized sum-product estimates. Its pinned version is due to Shmerkin [39]. The first nontrivial explicit dimensional exp onen t is due to Keleti and Shmerkin [28], who prop osed the nov el idea of scale- selection. A partial result based on estimates of Guth-Iosevich-Ou-W ang w as giv en b y the author [30], that w as later extended to all ev en dimensions b y W ang-Zheng [45]. The current record in the plane is due to Stull [44], who prov ed dim H E > 1 = ⇒ dim H ∆ y ( E ) ⩾ 3 4 , for some y ∈ E . His metho d is beyond the author’s understanding (as well as the recent w ork of Fiedler- Stull [19][20]), but according to Hong W ang (p ersonal communication), she and Shmerkin understand the k ey idea of Stull [44] and are able to obtain the same dimensional expo- nen t 3 4 via incidence estimates. W e also refer to the recent work of Shmerkin and W ang [42] for partial results on the endp oin t case dim H E ⩾ d/ 2 for all d ⩾ 2. Recen tly , the less w ell-known Assouad dimension dim A has dra wn muc h attention. The problem on dim A ∆( E ) in terms of dim A E has been solv ed by F raser in the plane [21][22]. Though op en in general, the distance set problem is b etter understo o d on sets with regularit y . B´ ar´ an y [1], Orp onen [35] studied self-similar sets, F erguson-F raser-Sahlsten [18] studied self-affine sets, F raser-Pollicott [23] studied self-conformal sets. A break- through in b ypassing dynamical systems was Orp onen’s work [36] on Ahlfors-David regular sets, and the idea w as further developed by Shmerkin [40][41], Keleti-Shmerkin PINNED DIST ANCE SET AND MIZOHA T A-T AKEUCHI-TYPE ESTIMA TES 3 [28]. The final v ersion can b e stated as the follo wing: supp ose E , F are Borel sets in the plane, then dim H F > 1 , dim H E = dim P E > 1 = ⇒ dim H ∆ y ( E ) = 1 , for some y ∈ E , where dim P denotes the packing dimension (see Section 2 b elow for definition). In par- ticular, if E ⊂ R 2 has equal Hausdorff and packing dimension > 1, then dim H ∆ y ( E ) = 1 for some y ∈ E . “dim H ∆( E ) = 1” is exciting and already answ ers the original question of F alconer in [16], but it is still weak er than “p ositive Leb esgue measure”. There is very little discussion on Leb esgue measure of distance sets of regular sets in the literature. The only w ork we kno w is due to Iosevich and the author [26] on | ∆( A 1 × A 2 ) | with A 1 , A 2 ⊂ R , Ahlfors-Da vid regular, dim A 1 + dim A 2 > 4 3 − ϵ 0 . In this pap er, w e settle the regular case of the distance set problem in the plane. Theorem 1.1. Supp ose E , F ar e Bor el sets in the plane, dim H E > 1 , dim H E + dim H F > 2 , and F has e qual Hausdorff and p acking dimension. Then ther e exists y ∈ F such that the pinne d distanc e set ∆ y ( E ) := {| x − y | : x ∈ E } has p ositive L eb esgue me asur e. In p articular, if E ⊂ R 2 has e qual Hausdorff and p acking dimension > 1 , then | ∆ y ( E ) | > 0 for some y ∈ E . Notice that there exists E ⊂ R 2 , dim H E = dim P E = 1, while | ∆( E ) | = 0. F or example, one can tak e E = A × { 0 } with A := ∞ \ i =1 { a ∈ [0 , 1] : dist( a, q − 1 i Z ) < q − 1 − ϵ i i } , where q i +1 > q i i > 1 and ϵ i ↓ 0 such that 10 q − ϵ i i → 0. Then dim H A = dim P A = 1 (see, for example, Section 8.5 in [17] for dim H A ⩾ 1 − ϵ i , ∀ i , and it is w ell known that dim H A ⩽ dim P A ⩽ 1). Therefore dim H E = dim P E = 1, while | ∆( E ) | = | A − A | ⩽ 10 q − ϵ i i → 0 . The main ingredients of the pro of of Theorem 1.1 consist of a multi-scale Go o d-Bad decomp osition and a m ulti-scale Mizohata-T akeuc hi-t yp e estimate with arbitrary small p o wer-loss. The single-scale Go o d-Bad decomp osition w as prop osed by Guth-Iosevich - Ou-W ang [25]. Our multi-scale Go o d-Bad decomp osition is inspired by them, but not a direct generalization (see the discussion in Section 3.2). As a result of the difference, it is the first time in recen t works that dim H F ⩽ 1 is allow ed for | ∆ y ( E ) | > 0 or dim ∆ y ( E ) = 1, y ∈ F . Also, unlike the refined decoupling inequality in [25], there is no deep L p theory in this pap er. W e develop an L 2 scale-reduction argumen t that has its own in terest (see Section 4). It also leav es plent y of ro om for more to ols to play a role in this framew ork. 1.2. Mizohata-T ak euchi conjecture. Let Σ b e a compact C 2 h yp ersurface in R d with surface measure σ . The Mizohata-T akeuc hi conjecture arose in the study of disp ersive PDE from 1970s. It states that, for ev ery non-negative w eight w on R d , Z R d | d f dσ ( x ) | 2 w ( x ) dx ≲ sup T w ( T ) Z Σ | f | 2 dσ, 4 BOCHEN LIU where w ( T ) := R T w and the supremum is taken o ver all 1-neigh b orho o ds of doubly- infinite lines. When Σ is a h yp erplane, it simply follo ws from Planc herel. On the other hand, unfortunately , it has b een pro ved false on every hypersurface that is not part of a h yp erplane, by a recent delicate example of Cairo [4]. F or more information on the history , related works, and discussion on the Mizohata-T ak euchi conjecture, we refer to the inspiring talk of Guth [24], comprehensive in tro ductions of Carb ery-Iliop oulou- W ang [7], Cairo [4], Cairo-Zhang [5], as w ell as recent w orks of Carb ery-H¨ anninen- V aldimarsson [6], Carb ery-Li-P ang-Y ung [8], Mulherk ar [34], and references therein. One wa y to describ e how far it is from b eing true is to consider its lo cal version: Z B R | d f dσ ( x ) | 2 w ( x ) dx ⩽ C R α · sup T w ( T ) Z Σ | f | 2 dσ. (1.1) Though the original Mizohata-T akeuc hi conjecture fails in general, it is an open problem that whether (1.1) holds for ev ery α > 0, called the lo cal Mizohata-T ak euchi conjecture. In [7], Carb ery , Iliopoulou and W ang pro ved (1.1) for every α > d − 1 d +1 on strictly conv ex C 2 h yp ersurfaces. Whether d − 1 d +1 is optimal is not known, but Guth [24] p ointed out that a R d − 1 d +1 − ϵ -p o wer-loss is inevitable under the usual “w a ve pack et decomp osition axioms”. See [7] for a detailed discussion on Guth’s outline. In a v ery recen t w ork of Cairo-Zhang [5], a family of coun terexamples are constructed, that in particular sho ws the exp onen t of Carb ery-Iliop oulou-W ang is sharp for man y strictly con vex C 2 h yp ersurfaces up to the endp oin t. But whether it is sharp for sp ecific surfaces, even for familiar ones like the parab oloid or the sphere, is still unkno wn. W e call (1.1) “single-scale” as all tub es under consideration hav e the same size. In this pap er, w e find that some multi-scale Mizohata-T ak euchi-t yp e estimates hav e no p o wer-loss, up to an arbitrary small constant δ > 0. The one we need for Theorem 1.1 is quite complicated to state. Instead, w e state the following bab y version in the in tro duction. This observ ation is also where this pro ject started. F rom now w e only consider the case Σ = S d − 1 , b ecause our motiv ation is on the distance set problem. The pro of also works on more general surfaces for asso ciated results. Theorem 1.2. L et σ denote the surfac e me asur e on S d − 1 . Supp ose w is a non-ne gative weight on B R ⊂ R d , 0 < δ ≪ 1 / 2 is a c onstant, and R δ = r 0 < · · · < r M = R is a se quenc e with r j +1 ⩽ r 2 j . Then w ( B R ) − 1 Z B R | d f dσ | 2 w ≲ d,δ,M R C δ     M − 1 Y j =0 sup ∀ Q j +1 ∈D r j +1 ∀ T j ∈ T r j ,r j +1 w Q j +1 ( T j )     ∥ f ∥ 2 L 2 ( S d − 1 ) , wher e D r j +1 denotes the c ol le ction of almost disjoint r j +1 -cub es, T r j ,r j +1 denotes the c ol le ction of r j × · · · × r j × r j +1 tub es, w ( B R ) := R B R w , w Q ( T ) := m ( Q ) − 1 R T ∩ Q w , and C > 0 is a c onstant indep endent in R, M , δ . When w = χ B R , this result b ecomes the trivial estimate ∥ d f dσ ∥ 2 L 2 ( B R ) ≲ d,δ R C δ · R ∥ f ∥ 2 L 2 ( S d − 1 ) , PINNED DIST ANCE SET AND MIZOHA T A-T AKEUCHI-TYPE ESTIMA TES 5 whic h is optimal up to the factor R C δ . The constrain r j +1 ⩽ r 2 j is due to tec hnical reasons. This is also why we need some regularity on the pin set F in Theorem 1.1. I w onder if it is necessary . A heuristic pro of of Theorem 1.2 is given in Section 4.1 to illustrate ideas. A detailed discussion on the v ersion for Theorem 1.1 is giv en in Section 4.2. Notation. W e list notation used in this pap er to mak e it easier for readers to recall. X ≲ Y means X ⩽ C Y for some constan t C > 0; X ≲ δ Y means X ⩽ C δ Y for some constan t C δ > 0 dep ending on δ > 0. W e omit the subindex δ in the pro of b ecause δ > 0 is a fixed constant. The constan t C > 0 in R C δ ma y v ary from line to line but is indep enden t in R and δ . All n umbers in this pap er can b e assumed dy adic. RapDec( R ) means it is ⩽ C N R − N for arbitrary large N > 0, called a negligible error in the pro of. D r denotes the collection of almost disjoint half-op en-half-closed r -cub es that co ver R d . W e ma y assume all cub es in this pap er are dy adic. F or a cub e Q , denote b y 2 Q its dilation by factor 2 from the center. Once the collection D r is clear, denote by Q ( y ) the Q ∈ D r that contains y . Let N ( E , r ) denote the smallest n umber of r -balls needed to cov er E . Let m denote the Leb esgue measure. W e also use | A | := m ( A ) to denote the Leb esgue measure of a set A . Let σ r denote the normalized surface measure on r S d − 1 . Also denote σ = σ 1 . F or a measure ν , denote b y ν Q := ν ( Q ) − 1 ν | Q the normalization of ν restricted on Q . b f ( ξ ) := R e − 2 π ix · ξ f ( x ) dx is the F ourier transform and f ∨ ( ξ ) := R e 2 π ix · ξ f ( x ) dx is the in verse F ourier transform. The direction of a tub e T , denoted b y θ ( T ), is the direction of its long axis. W e say a tub e T is parallel to a cap in S d − 1 if its direction matches the center of the cap. Denote b y 2 T its dilation by factor 2 from the cen ter of T . F or x  = y , let l x,y denote the line connecting x, y . Denote b y T r 1 × r 2 ( x ; y ) the r 1 × r 2 - tub e centered at x of direction l x,y . Organization. This pap er is organized as follo ws. In Section 2, we review differen t dimensions and some standard techniques in mo dern harmonic analysis. In Section 3, w e first review the single-scale Go o d-Bad decomp osition of Guth-Iosevic h-Ou-W ang, then introduce our m ulti-scale Go o d-Bad decomp osition and pro ve the L 1 -estimate of our bad part. In Section 4, we first giv e a heuristic pro of of Theorem 1.2, then give a detailed pro of of the L 2 -estimate of our go o d part. In Section 5, we use our estimates on go o d and bad parts to pro ve Theorem 1.1. Ac knowledgemen t. The author w ould lik e to thank T uomas Orp onen for comments and suggestions on an earlier draft that help ed improv e the man uscript. 6 BOCHEN LIU 2. Preliminaries There are t w o equiv alen t w a ys to define Hausdorff dimension of sets E ⊂ R d in terms of measures: dim H E = sup s { s : ∃ a finite Borel measure µ on E with µ ( B ( x, r )) < r s , ∀ x ∈ R d , r > 0 } = sup s { s : ∃ a finite Borel measure µ on E with I s ( µ ) < ∞} , where I s ( µ ) is the s -dimension energy of µ defined b y I s ( µ ) := Z Z | x − y | − s dµ ( x ) dµ ( y ) = c s,d Z | b µ ( ξ ) | 2 | ξ | − d + s dξ . A finite Borel measure µ satisfying µ ( B ( x, r )) ≲ r s , ∀ x ∈ R d , r > 0 , is called a F rostman measure of dimension s . Let N ( E , r ) denote the smallest n umber of r -balls needed to cov er E . The upp er Mink owski dimension (also called the upp er b ox-coun ting dimension) is defined by dim M E := lim sup r → 0 log N ( E , r ) log 1 /r . The packing dimension of E , denoted b y dim P E , is defined b y dim P E := inf { sup i dim M E i : E = ∞ [ i =1 E i , E i is b ounded } . In particular, for ev ery finite Borel measure µ on E and ev ery ϵ > 0, there exists E i ⊂ E suc h that dim M E i < dim P E + ϵ and µ ( E i ) > 0. It is easy to v erify that dim H E ⩽ dim P E ⩽ dim M E , and dim H E = dim P E if E is Ahlfors-Da vid regular (not vice versa). F or details and more discussion on their relations, we refer to Mattila’s b o oks [33] as a reference. W e will need several standard tec hniques in modern harmonic analysis. W e only state the version for our use. The pro ofs are not hard, so w e omit details and refer to, for example, Guth’s lecture notes (see his homepage) and Demeter’s b o ok [9], as references. Throughout this pap er, D r denotes the collection of almost disjoint half-op en-half- closed r -cub es that co ver R d , R ϵ Q denotes the dilation b y factor R ϵ from the cen ter of Q , m denotes the Leb esgue measure, and m Q := m ( Q ) − 1 m | Q denotes the normalized Leb esgure measure on Q . • (Lo calization) Supp ose supp b f ⊂ B R and ν is a finite Borel measure. Then for all ϵ > 0, Z | f | 2 dν ≲ ϵ R ϵ X Q ∈D R − 1 ν ( R ϵ Q ) Z | f | 2 dm R ϵ Q + RapDec( R ) ∥ f ∥ 2 L 2 ( m ) . • (Lo cal L 2 -orthogonalit y) Supp ose { f i } is a family of functions with F ourier trans- form supp orted on R -balls of bounded ov erlapping. Then on eac h R − 1 -ball B R − 1 , Z | X f i | 2 dm B R − 1 ≲ ϵ R ϵ X Z | f i | 2 dm R ϵ B R − 1 + RapDec( R ) X ∥ f i ∥ 2 L 2 ( m ) . PINNED DIST ANCE SET AND MIZOHA T A-T AKEUCHI-TYPE ESTIMA TES 7 • (Lo cal constancy) Supp ose b f is supp orted in a R 1 × · · · × R d rectangle T in R d , then for every dual rectangle T ∗ , that is a R − 1 1 × · · · × R − 1 d rectangle whose R − 1 j -axis is parallel to the R j -axis of T , we ha ve, for all ϵ > 0, ∥ f ∥ 2 L ∞ ( T ∗ ) ≲ ϵ R ϵ ∥ f ∥ 2 L 2 ( m R ϵ T ∗ ) + RapDec( R ) ∥ f ∥ 2 L 2 ( m ) . 3. A mul ti-scale Good-Bad decomposition 3.1. The single-scale Go o d-Bad decomp osition of Guth-Iosevich-Ou-W ang. F or nearly 30 y ears, the L 2 -metho d has b een the most efficien t approach to the F alconer distance conjecture, until Guth-Iosevic h-Ou-W ang [25] p ointed out that it cannot b eat W olff ’s partial result in the plane [46]. T o go further, they prop osed the no vel Go o d-Bad decomp osition as follows. Let R 0 b e a large dyadic n umber and R i := 2 i R 0 . F or each i ⩾ 1, cov er the unit circle b y R − 1 / 2 i -caps θ , and for each θ , find a R 1 / 2 i × 10 R i rectangle parallel to θ such that the union of these rectangles co vers the ann ulus | ξ | ≈ R i . Let { ψ i,θ } be a partition of unity sub ordinate to this co ver and write 1 = ψ 0 + X i ⩾ 1 X θ : R − 1 / 2 i -caps ψ i,θ . (3.1) Let δ > 0 b e a small fixed constant. F or each ( i, θ ), co ver the unit ball B 1 with R − 1 / 2+ δ i × 1 tub es T parallel to θ . Denote T i,θ as the collection of these tub es, and let η T b e a partition of unit y sub ordinate to this cov er suc h that 1 = X T ∈ T i,θ η T on B 2 . Also denote T i := [ θ : R − 1 / 2 i -caps T i,θ . (3.2) No w, supp ose µ is a finite Borel measure on E ⊂ B 1 and ν is a finite Borel measure on F ⊂ B 1 of disjoint supports. W e say a tub e T ∈ T i is bad if ν ( T ) ⩾ R − 1 / 2+100 δ i , and go o d otherwise. Define M 0 µ := ( ψ 0 b µ ) ∨ , M T µ := η T ( ψ i,θ b µ ) ∨ , if T ∈ T i,θ , i ⩾ 1 , (3.3) and µ g ood := M 0 µ + X i ⩾ 1 X T ∈ T i , g ood M T µ. If one can sho w that ∥ d y ∗ ( µ g ood ) − d y ∗ ( µ )) ∥ L 1 ( R × ν ) = ∥ X i ⩾ 1 X T ∈ T i , bad d y ∗ ( M T µ ) ∥ L 1 ( R × ν ) < 1 / 100 (3.4) and ∥ d y ∗ ( µ g ood ) ∥ L 2 ( R × ν ) < ∞ , (3.5) 8 BOCHEN LIU then 1 = Z Z ∆ y ( E ) d y ∗ ( µ ) ! dν ( y ) ⩽ ∥ d y ∗ ( µ g ood ) − d y ∗ ( µ )) ∥ L 1 +( sup y ∈ supp ν | ∆ y ( E ) | 1 / 2 ) ∥ d y ∗ ( µ g ood ) ∥ L 2 implies that | ∆ y ( E ) | > 0 , for some y ∈ F . In [25], under the assumption dim H F > 1 and 3 dim H E + dim H F > 5, the bad part estimate (3.4) follows from a radial pro jection theorem of Orp onen [37]: supp ose π x ( y ) := y − x | y − x | and t ∈ (1 , 2), then there exists p > 1 suc h that ∥ π x ∗ ν ∥ L p ( S 1 × µ ) ≲ I t ( ν ) 1 2 I 2 − t ( µ ) 1 2 p ; (3.6) and the go o d part estimate (3.5) follo ws from a remark able refined decoupling inequality . W e p oint out that the constrain t dim H F > 1 in their result arises from the radial pro jection estimate (3.6). W e shall need the following lemma from [25]. Lemma 3.1 (Lemma 3.5 in [25]) . F or e ach y ∈ supp ν , ∥ d y ∗ ( µ g ood ) − d y ∗ ( µ ) ∥ L 1 ( R ) = ∥ X i ⩾ 1 X T ∈ T i , bad d y ∗ ( M T µ ) ∥ L 1 ( R ) ⩽ X i ⩾ 1 X T ∈ T i ,bad ∥ d y ∗ ( M T µ ) ∥ L 1 ( R ) ≲ X i ⩾ 1 µ ( B ad i ( y )) + R apDe c ( R 0 ) , wher e B ad i ( y ) := [ T ∈ T i , bad, y ∈ 2 T 2 T . 3.2. A m ulti-scale Go o d-Bad decomp osition. W e call the Goo d-Bad decompo- sition of Guth-Iosevic h-Ou-W ang “single-scale”, b ecause only R − 1 / 2+ δ i × 1 tubes are considered for each i ⩾ 1. In this pap er, w e prop ose a m ulti-scale Go o d-Bad decomp osition. Though inspired b y Guth-Iosevic h-Ou-W ang, our Goo d-Bad decomp osition is not a direct generalization of theirs. Heavy tub es are remo ved in differen t w a ys. In [25], Orp onen’s L p radial pro jection estimate (3.6) is used to remo ve hea vy tub es on F , while in this pap er, it is used to remov e heavy tubes on E , then an L 2 -tec hnique remov es heavy tubes on F . See the pro of of Prop osition 3.3 b elow for details. This is also why dim H F ⩽ 1 is allow ed in our Theorem 1.1. Fix i ⩾ 1, let R := R i b e as in the previous subsection, and δ > 0 be a small constan t that will b e clarified later. Consider the sequence r j := R 2 j δ , j = 0 , . . . , log 2 1 /δ. (3.7) Divide the unit sphere in to r − 1 0 -caps θ 0 , then divide eac h θ 0 in to r − 1 1 -caps θ 1 ⊂ θ 0 ,..., and finally into r − 1 log 1 /δ − 1 = R − 1 / 2 -caps θ log 1 /δ − 1 . W e may assume that { θ log 1 /δ − 1 } coin- cides with { θ } in the definition of T i in (3.2). PINNED DIST ANCE SET AND MIZOHA T A-T AKEUCHI-TYPE ESTIMA TES 9 F or eac h cub e Q with ν ( Q )  = 0, let ν Q := ν ( Q ) − 1 ν | Q denote the normalization of ν restricted on Q . Below we only consider cub es with nonzero measure. F or eac h r − 1 j -cap θ j and each dyadic r j /R -cub e Q j ∈ D r j /R , w e sa y θ j is bad for Q j if the r j /R × r j +1 /R tub e T r j /R × r j +1 /R cen tered at the center of Q j of direction θ j satisfies ν R 2( j +2) δ 2 Q j +1 ( R δ T r j /R × r j +1 /R ) > R 10 δ · ( r j /r j +1 ) min { t, 1 } , (3.8) where Q j +1 ∈ D r j +1 /R is the dyadic r j +1 /R -cub e con taining Q j . F or eac h Q j ∈ D r j /R , denote Θ j ( Q j ) := { θ j : r − 1 j -caps, bad for Q j } . (3.9) More generally , for Q j ′ ∈ D r j ′ /R , Q j ∈ D r j /R , Q j ′ ⊂ Q j , j ′ ⩽ j , define Θ j ( Q j ′ ) := Θ j ( Q j ) . (3.10) Recall T i from (3.2). F or eac h dyadic R − 1+ δ i -cub e Q ∈ D r 0 /R and j = 0 , . . . , log δ − 1 − 1, denote T i,Q,j := { T ∈ T i : 2 T ∩ Q  = ∅ and ∃ θ j ∈ Θ j ( Q ) with θ ( T ) ∈ θ j } . (3.11) Then let T i,Q := [ 0 ⩽ j ⩽ log δ − 1 − 1 T i,Q,j . (3.12) No w we are ready to decomp ose µ in to go o d and bad parts. Recall the definition of M 0 µ, M T µ from (3.3). First, for each i ⩾ 1 and eac h dyadic R − 1+ δ i -cub e Q ∈ D r 0 /R , define µ g ood,i,Q := X T ∈ T i \ T i,Q M T µ. (3.13) Then, let i ⩾ 1 v ary , and for each y ∈ supp ν , define µ g ood,y := M 0 µ + X i ⩾ 1 µ g ood,i,Q ( y ) , (3.14) where Q ( y ) in each µ g ood,i,Q ( y ) denotes the dyadic R − 1+ δ i -cub e containing y . Compared with the single-scale case, our µ g ood,y dep ends on y . This is because the asso ciated bad caps are different for differen t y ∈ supp ν . It brings additional difficulties on the L 2 -estimate of the go o d part in the next section. On the other hand, for the L 1 -estimate of the bad part, Lemma 3.1 still applies b ecause it is a point wise estimate. Consequen tly , for each y ∈ supp ν , ∥ d y ∗ ( µ g ood,y ) − d y ∗ ( µ ) ∥ L 1 ( R ) = ∥ X i ⩾ 1 X T ∈ T i,Q ( y ) d y ∗ ( M T µ ) ∥ L 1 ( R ) ⩽ X i ⩾ 1 X T ∈ T i,Q ( y ) ∥ d y ∗ ( M T µ ) ∥ L 1 ( R ) ≲ X i ⩾ 1 µ ( B ad i ( y )) + RapDec( R 0 ) , 10 BOCHEN LIU where, this time, B ad i ( y ) := [ T ∈ T i,Q ( y ) 2 T . Then, with B ad i := { ( x, y ) ∈ supp µ × supp ν : x ∈ B ad i ( y ) } = { ( x, y ) ∈ supp µ × supp ν : ∃ T ∈ T i,Q ( y ) , x ∈ 2 T } , (3.15) it follows that Z ∥ d y ∗ ( µ g ood,y ) − d y ∗ ( µ )) ∥ L 1 ( R ) dν ( y ) ≲ X i ⩾ 1 Z µ ( B ad i ( y )) dν ( y ) + RapDec( R 0 ) = X i ⩾ 1 µ × ν ( B ad i ) + RapDec( R 0 ) . Then w e need the follo wing geometric observ ation on pairs ( x, y ) ∈ B ad i : by our definition of “bad” in (3.8)-(3.12) and (3.15), there exist 0 ⩽ j ⩽ log 1 /δ − 1 and T ∈ T i suc h that x, y ∈ 10 T and θ ( T ) lies a r − 1 j -cap θ j bad for Q j ( y ) ∈ D r j /R . Therefore, if w e use T r j /R × r j +1 /R ( y ; x ) to denote the r j /R × r j +1 /R tub e centered at y parallel to the line l x,y connecting x, y , then R 2 δ T r j /R × r j +1 /R ( y ; x ) ⊃ R δ T r j /R × r j +1 /R , where T r j /R × r j +1 /R is the r j /R × r j +1 /R -tub e centered at the cen ter of Q j ( y ) of direction θ j from (3.8). Consequently , ν R 2( j +2) δ 2 Q j +1 ( R 2 δ T r j /R × r j +1 /R ( y ; x )) > R 10 δ · ( r j /r j +1 ) min { t, 1 } , and therefore the set B ad i is a subset of { ( x, y ) ∈ E × F : ∃ j, ν R 2( j +2) δ 2 Q j +1 ( y ) ( R 2 δ T r j /R × r j +1 /R ( y ; x )) > R 10 δ · ( r j /r j +1 ) min { t, 1 } } . Moreo ver, B ad i is a subset of the construction in Prop osition 3.3 b elo w with R = R i (it seems easier to compare their complemen t). Hence, if µ has finite s -energy with s > 1 and ν is F rostman of dimension t > 2 − s with dim M supp ν < t + δ 2 , we ha v e µ × ν ( B ad i ) ≲ (log R ) 2 R − δ 3 (3.16) and finally conclude the follo wing. Prop osition 3.2. Supp ose µ, ν ar e pr ob ability me asur es on the unit b al l with disjoint supp ort, I s ( µ ) < ∞ for some s > 1 , and ν is F r ostman of dimension t > 2 − s with dim M supp ν < t + δ 2 . Then, when R 0 is lar ge enough and δ is smal l enough in terms of s, t , Z ∥ d y ∗ ( µ g ood,y ) − d y ∗ ( µ )) ∥ L 1 ( R ) dν ( y ) < 1 / 100 . Here is a more general result than what w e need for (3.16). Prop osition 3.3. Supp ose µ, ν ar e pr ob ability me asur es on [0 , 1) 2 with disjoint supp ort, I s ( µ ) < ∞ for some s > 1 and ν is F r ostman of dimension t > 2 − s with dim M supp ν < t + δ 2 . Then, for al l R > 1 and δ > 0 smal l enough in terms of s, t , ther e exists a subset B ad ⊂ [0 , 1) 2 such that • µ × ν ( B ad ) ≲ (log R ) 2 R − δ 3 , and PINNED DIST ANCE SET AND MIZOHA T A-T AKEUCHI-TYPE ESTIMA TES 11 • for al l p airs of p oints ( x, y ) / ∈ B ad and al l dyadic numb ers r , r ′ with R − 1+ δ ⩽ R δ r ′ ⩽ r ⩽ 1 , the r ′ × r tub e T r ′ × r ( y ; x ) c enter e d at y p ar al lel to the line l x,y c onne cting x, y satisfies ν Q ( T r ′ × r ( y ; x )) < R δ · ( r ′ /r ) min { t, 1 } , wher e Q = Q ( y ) ∈ D r is the dyadic r -cub e c ontaining y . Pr o of of Pr op osition 3.3. Fix r ′ , r . Step 1. W e first consider pairs ( x, y ) ∈ supp µ × supp ν with µ ( T r/r ′ × 1 ( x ; y )) > R δ 2 · r ′ /r , and denote the set of suc h pairs by B ad r,r ′ ,µ , whic h is an op en subset of [0 , 1) 2 . The estimate on µ × ν ( B ad r,r ′ ,µ ) is the same as Lemma 3.6 in [25], just with µ, ν sw app ed. W e giv e the pro of for completeness. F or each y ∈ supp ν , as dist( y , supp µ ) ≈ 1, one can co ver supp µ by ≈ r /r ′ man y r ′ /r × 1 tubes con taining y of b ounded o v erlapping. Denote this collections of tubes b y T y ,r ′ /r × 1 . Then, for eac h x with ( x, y ) ∈ B ad r,r ′ ,µ , there exists T ∈ T y ,r ′ /r × 1 suc h that T r/r ′ × 1 ( x ; y ) ⊂ 2 T . Because of b ounded o verlapping, there are ≲ R − δ 2 · r /r ′ man y T ∈ T y ,r ′ /r × 1 with µ (2 T ) > R δ 2 · r ′ /r . Consequen tly , for each y ∈ supp ν , { x : ( x, y ) ∈ B ad r,r ′ ,µ } is con tained in a union of ≲ R − δ 2 · r /r ′ man y r ′ /r × 1 tub es con taining y . Therefore, if w e denote π y ( x ) := y − x | y − x | as the radial pro jection, then µ × ν ( B ad r,r ′ ,µ ) = Z µ ( { x : ( x, y ) ∈ B ad r,r ′ ,µ } ) dν ( y ) = Z  Z π y ∗ µ  dν ( y ) , and for eac h y ∈ supp ν , the supp ort of π y ∗ ( µ ) is con tained in ≲ R − δ · r /r ′ man y r ′ /r - arcs in S 1 . Hence, since s > 1, by Orp onen’s radial pro jection estiamte (3.6) (with µ, ν sw app ed), there exists p > 1 suc h that µ × ν ( B ad r,r ′ ,µ ) ≲ R − δ 2 /p ′ · ∥ π y ∗ µ ∥ L p ( S 1 × ν ) ≲ R − δ 2 /p ′ I s ( µ ) 1 2 I 2 − s ( ν ) 1 2 p ≲ R − C δ 2 , where the constant C in the exp onent only dep ends on s, t . As a remark, the existence of some p > 1 is sufficien t here. W e refer to [31] for a discussion on the range of p . Step 2. Remov e those Q ∈ D r with ν ( Q ) < R − δ 2 r t . More precisely , as dim M supp ν < t + δ 2 , the cov ering n umber satisfies N (supp ν, r ) ≲ r − t − δ 2 , whic h implies # { Q ∈ D r : Q ∩ supp ν  = ∅} ≲ r − t − δ 2 . 12 BOCHEN LIU Therefore, X Q ∈D r , ν ( Q ) 0 is small enough in terms of s, t . Step 3. On [0 , 1) 2 \ B ad r,r ′ ,µ,ν , write µ × ν = X Q ∈D r ν ( Q ) · µ × ν Q . F or eac h Q ∈ D r and y ∈ Q , consider pairs ( x, y ) ∈ [0 , 1) × Q with ν Q ( T r ′ × r ( y ; x )) > R δ · ( r ′ /r ) min { t, 1 } , (3.17) and denote the set of these pairs b y B ad r,r ′ ,Q ⊂ ( B ad r,r ′ ,µ,ν ) c . Cho ose ≈ r /r ′ man y r ′ /r -separated directions { e i } ⊂ S 1 . F or eac h selected direction e i , cov er Q by ≈ r/r ′ man y r ′ × r tub es parallel to e i of b ounded ov erlapping. Denote the collection of all selected r ′ × r tub es by T Q,r ′ × r . Here comes the key observ ation in this step: for eac h ( x, y ) ∈ B ad r,r ′ ,Q , there exist T ∈ T Q,r ′ × r , an asso ciated r ′ /r × 1 tub e of the same cen tral line of T r ′ /r × 1 , denoted b y T r ′ /r × 1 ∼ T , suc h that T r ′ × r ( y ; x ) ⊂ 2 T and x ∈ T r ′ /r × 1 . This observ ation implies that, if we call T ∈ T Q,r ′ × r bad when ν (2 T ) > R δ · ( r ′ /r ) min { t, 1 } , then B ad r,r ′ ,Q ⊂ [ T ∈ T Q,r ′ × r , bad T r ′ /r × 1 ∼ T T r ′ /r × 1 × T . Since ( x, y ) / ∈ B ad r,r ′ ,µ , by our construction from Step 1, it follows that µ ( T r ′ /r × 1 ) < R δ 2 · r ′ /r . Therefore, µ × ν Q ( B ad r,r ′ ,Q ) ⩽ X T ∈ T Q,r ′ × r , bad T r ′ /r × 1 ∼ T µ ( T r ′ /r × 1 ) ν Q ( T ) ⩽ R δ 2 · r ′ /r X T ∈ T Q,r ′ × r , bad ν Q ( T ) = R δ 2 · r ′ /r X e i ∈ S 1 r ′ /r -separated X T ∈ T Q,r ′ × r T ∥ e i , bad ν Q ( T ) . PINNED DIST ANCE SET AND MIZOHA T A-T AKEUCHI-TYPE ESTIMA TES 13 As tub es in T Q,r ′ × r ha ve b ounded ov erlapping, b y (3.17), for eac h e i , the num b er of T , in the sum is ≲ R − δ ( r /r ′ ) min { t, 1 } . So, by Cauc hy-Sc h warz, ( µ × ν Q ( B ad r,r ′ ,Q )) 2 ≲ ( R δ 2 r ′ /r ) 2 · r /r ′ · R − δ ( r /r ′ ) min { t, 1 } X e i ∈ S 1 r ′ /r -separated X T ∈ T Q,r ′ × r T ∥ e i , bad ν Q ( T ) 2 ⩽ R 2 δ 2 − δ ( r ′ /r ) 1 − min { t, 1 } X T ∈ T Q,r ′ × r ν Q ( T ) 2 = R 2 δ 2 − δ ( r ′ /r ) 1 − min { t, 1 } Z Z X T ∈ T Q,r ′ × r χ T ( y ) χ T ( y ′ ) dν Q ( y ) dν Q ( y ′ ) . Fix y ′ and consider | y − y ′ | ≈ 2 − j with 2 − j ∈ [ r ′ , r ]. Then the n umber of T ∈ T Q,r ′ × r con taining b oth y , y ′ is ≲ 2 j r . Also, since ν is F rostman of dimension t and ν ( Q ) ⩾ R − δ 2 r t from Step 2, it follo ws that ν Q ( B ( y , 2 − j )) ≲ R δ 2 (2 j r ) − t . T ogether w e hav e ( µ × ν Q ( B ad r,r ′ ,Q )) 2 ≲ R 4 δ 2 − δ ( r ′ /r ) 1 − min { t, 1 }   ( r ′ /r ) t + X j :2 − j ∈ [ r ′ ,r ] 2 j r · (2 j r ) − t   ≲ R 4 δ 2 − δ , whic h is ≲ R − δ 2 when δ < 1 / 10. Step 3. Finally tak e B ad := [ r ′ ,r : dy adic R − 1+ δ ⩽ R δ r ′ ⩽ r ⩽ 1 [ Q ∈D r B ad r,r ′ ,Q ∪ B ad r,r ′ ,µ,ν . F rom previous steps, µ × ν ( B ad ) ≲ X r,r ′ R − δ 3 + X Q ∈D r ν ( Q ) R − δ 3 ! ≲ (log R ) 2 R − δ 3 when δ > 0 is small enough in terms of s, t , and all required conditions are satisfied. □ 4. Mul ti-scale Mizoha t a-T akeuchi-type estima tes 4.1. Pro of of Theorem 1.2. In this subsection, w e giv e a heuristic pro of of Theorem 1.2 that ignores man y δ -p ow er-loss. W e use the symbol ⪅ for this ignorance. A rigorous argumen t on the same idea will b e given in the next subsection. Let ψ ∈ C ∞ 0 b e non-negative with b ψ  = 0 on the unit ball. Denote ψ R ( · ) := R d ψ ( R · ). Then, as the in tegral is o ver B R , Z B R | d f dσ | 2 w ≲ Z B R | \ ( f dσ ) ∗ ψ R | 2 w := Z B R | F | 2 w . F or eac h 0 ⩽ j ⩽ log 1 /δ − 1, divide the unit sphere in to r − 1 j -caps θ j and let ψ θ j b e a partition of unity sub ordinate to this co ver. Denote F θ j := \ ( f ψ θ j dσ ) ∗ ψ R . 14 BOCHEN LIU First, by the lo calization, we ma y assume w is a constant on eac h r 0 -cub e, namely , ∥ F ∥ 2 L 2 ( w ) ⪅ X Q 0 ∈D r 0 w ( Q 0 ) Z | F | 2 dm Q 0 . (4.1) Second, as the supp orts of c F θ 0 lie in almost disjoin t r − 1 0 -balls, the lo cal L 2 -orthogonalit y b et ween F θ 0 s on each Q 0 ∈ D r 0 implies Z | F | 2 dm Q 0 ⪅ X θ 0 : r − 1 0 -caps Z | F θ 0 | 2 dm Q 0 . (4.2) Third, as c F θ 0 is supported on a r − 1 0 × · · · × r − 1 0 × r − 1 1 rectangle perp endicular to θ 0 , the function F θ 0 is lo cally a constan t on eac h r 0 × · · · × r 0 × r 1 tub e parallel to θ 0 . Therefore, if we co ver Q 1 ∈ D r 1 b y r 0 × · · · × r 0 × r 1 tub es T 0 parallel to θ 0 with b ounded o verlapping, then X Q 0 ⊂ Q 1 w ( Q 0 ) Z | F θ 0 | 2 dm Q 0 = X T 0 ∥ θ 0 X Q 0 ⊂ T 0 w ( Q 0 ) Z | F θ 0 | 2 dm Q 0 ⪅ X T 0 ∥ θ 0 X Q 0 ⊂ T 0 w ( Q 0 ) ∥ F θ 0 ∥ 2 L ∞ ( T 0 ) = X T 0 ∥ θ 0 w ( T 0 ) ∥ F θ 0 ∥ 2 L ∞ ( T 0 ) ⪅ w ( Q 1 ) X T 0 ∥ θ 0 w Q 1 ( T 0 ) m Q 1 ( T 0 ) Z T 0 | F θ 0 | 2 dm Q 1 , whic h is, b ecause of m Q 1 ( T 0 ) ≈ ( r 1 /r 0 ) d − 1 , ⪅ ( r 0 /r 1 ) d − 1    sup ∀ Q 1 ∈D r 1 ∀ T 0 ∈ T r 0 ,r 1 w Q 1 ( T 0 )    w ( Q 1 ) Z | F θ 0 | 2 dm Q 1 . T ogether with (4.1) and (4.2), w e obtain ∥ F ∥ 2 L 2 ( w ) ⪅ X Q 0 ∈D r 0 w ( Q 0 ) Z | F | 2 dm Q 0 ⪅ ( r 1 /r 0 ) d − 1    sup ∀ Q 1 ∈D r 1 ∀ T 0 ∈ T r 0 ,r 1 w Q 1 ( T 0 )    X θ 0 X Q 1 ∈D r 1 w ( Q 1 ) Z | F θ 0 | 2 dm Q 1 . By iterating this pro cess, we end up with ∥ F ∥ 2 L 2 ( w ) ⪅ R d − 1     M − 1 Y j =0 sup ∀ Q j +1 ∈D r j +1 ∀ T j ∈ T r j ,r j +1 w Q j +1 ( T j )     w ( B R ) X θ : R − 1 / 2 -caps Z | F θ | 2 dm B R , PINNED DIST ANCE SET AND MIZOHA T A-T AKEUCHI-TYPE ESTIMA TES 15 and Theorem 1.2 follo ws by the standard “Plancherel & Cauc hy-Sc hw arz” argumen t: X θ : R − 1 / 2 -caps Z | F θ | 2 dm B R ≲ R − d X θ : R − 1 / 2 -caps Z | \ ( f ψ θ dσ ) ∗ ψ R | 2 dm = R − d X θ : R − 1 / 2 -caps Z | ( f ψ θ dσ ) ∗ ψ R | 2 dm ≲ R − ( d − 1) X θ : R − 1 / 2 -caps Z | f ψ θ ( σ ) | 2  Z | ψ R ( x − σ ) | dx  dσ ≲ R − ( d − 1) Z | f | 2 dσ. 4.2. L 2 estimates of the go o d part. Prop osition 4.1. Supp ose µ, ν ar e pr ob ability me asur es on the unit b al l with disjoint supp ort, I s ( µ ) < ∞ for some s > 1 , and ν is F r ostman of dimension t > 2 − s with dim M supp ν < t + δ 2 . Then, with µ g ood,y define d in (3.14) , when δ > 0 is smal l enough in terms of s, t , Z ∥ d y ∗ ( µ g ood,y ) ∥ 2 L 2 ( R ) dν ( y ) < ∞ . By the observ ation in [29], for each y ∈ supp ν , ∥ d y ∗ ( µ g ood,y ) ∥ 2 L 2 ( R ) = Z ∞ 0 | µ g ood,y ∗ σ t ( y ) | 2 t 2 dt, where σ t denotes the normalized arc-length measure on tS 1 . This is ≲ Z ∞ 0 | µ g ood,y ∗ σ t ( y ) | 2 tdt b ecause µ g ood,y is essentially supp orted near supp µ , a wa y from y ∈ supp ν (see Lemma 5.2 in [25]). Then, b y the L 2 -iden tity pro ved in [29], for every y ∈ R 2 , Z ∞ 0 | µ g ood,y ∗ σ t ( y ) | 2 tdt = Z ∞ 0 | µ g ood,y ∗ b σ r ( y ) | 2 r dr . By the definition of µ g ood,i,Q ( y ) , µ g ood,y in (3.13), (3.14), it suffices to show Z ∞ 0 | µ g ood,i,Q ( y ) ∗ b σ r ( y ) | 2 r dr ≈ Z r ≈ R i | µ g ood,i,Q ( y ) ∗ b σ r ( y ) | 2 r dr + RapDec( R i ) is summable in i ⩾ 1, where the righ t hand side follo ws b ecause the F ourier supp ort of µ g ood,i,Q ( y ) lies in the ann ulus | ξ | ≈ R i . Notice that the dep endence on y in µ g ood,y has not y et made any difference b ecause so far everything is p oint wise. No w w e fix R ≈ R i and consider the in tegral with resp ect to dν ( y ): Z | µ g ood,i,Q ( y ) ∗ c σ R ( y ) | 2 dν ( y ) ≲ Z | (( \ µ g ood,i,Q ( y ) dσ R ) ∗ ψ ) ∨ ( y ) | 2 dν ( y ) , (4.3) where ψ ∈ C ∞ 0 is non-negativ e with b ψ  = 0 on the unit ball, and the right hand side follo ws b ecause supp ν ⊂ B 1 . T o deal with the dep endence on Q ( y ), a careful analysis on bad caps asso ciated with differen t y is required. 16 BOCHEN LIU Let r j b e as in (3.7) and recall the definition of Θ j in (3.9), (3.10). Notice that Θ j ( Q j 1 ) = Θ j ( Q j 2 ) = Θ j ( Q j ) (4.4) for all Q j 1 ∈ D r j 1 /R , Q j 2 ∈ D r j 2 /R con tained in the same Q j ∈ D r j /R , ∀ j 1 , j 2 ⩽ j . F or 0 ⩽ k < j ⩽ log 1 /δ − 1 and Q k ∈ D r k /R , let Θ = j ( Q k ) := { θ j ∈ Θ j ( Q k ) : disjoint with all caps in Θ k ( Q k ) , . . . , Θ j − 1 ( Q k ) } . (4.5) No w w e can rewrite the integrand in (4.3) to the follo wing. Lemma 4.2. R e c al l ψ i,θ define d in (3.1) , and let F := (( b µ dσ R ) ∗ ψ ) ∨ , F θ := (( ψ i,θ b µ dσ R ) ∗ ψ ) ∨ , F θ j := X θ ⊂ θ j F θ . Then for every Q ∈ D r 0 /R and y ∈ Q , (( \ µ g ood,i,Q dσ R ) ∗ ψ ) ∨ ( y ) = F ( y ) − X θ 0 ∈ Θ 0 ( Q ) F θ 0 ( y ) − log 1 /δ − 1 X j =1 X θ j ∈ Θ = j ( Q ) F θ j ( y ) + R apDe c ( R ) . Pr o of of L emma 4.2. First, by the definition of T i,Q,j , T i,Q , µ g ood,i,Q in (3.11)-(3.13), µ g ood,i,Q = X T ∈ T i M T µ − X T ∈ T i,Q, 0 M T µ − log 1 /δ − 1 X j =1 X T ∈ T i,Q,j θ ( T ) ∈ Θ = j ( Q ) M T µ, where θ ( T ) ∈ Θ = j ( Q ) means θ ( T ) lies in some θ j ∈ Θ j ( Q ). Then, by the definition of T i,Q,j in (3.11), Lemma 4.2 follo ws from a basic property of wa ve pac kets: ∥ (( [ M T µ dσ R ) ∗ ψ ) ∨ ∥ L ∞ ( Q ) = RapDec( R ) , given 2 T ∩ Q = ∅ . (4.6) More precisely , b y the definition of M T µ in (3.3), one can write out (( [ M T µ dσ R ) ∗ ψ ) ∨ ( y ) = ψ ∨ ( y ) Z  Z S 1 Z e 2 π i ( y · Rσ − x · Rσ + x · ξ ) η T ( x ) dxdσ  ψ i,θ ( ξ ) b µ ( ξ ) dξ , then integration by parts on x sho ws that it is RapDec( R ) unless σ lies in the R − 1 / 2 - neigh b orho o d of θ , and finally , b ecause the angle b et ween θ and l x,y , ∀ x ∈ T , ∀ y ∈ Q is > R − 1 / 2+ δ , in tegration b y parts on σ under lo cal co ordinates concludes (4.6). W e refer to [43], Chapter VI I I, for details of the second integration by parts, and [9] as a reference for wa v e pac kets. □ F or eac h 0 ⩽ k ⩽ log 1 /δ , eac h r k /R -cub e Q k ∈ D r k /R , and each r − 1 k − 1 -cap θ k − 1 , let F Q k ,θ k − 1 := F θ k − 1 − X θ k ∈ Θ k ( Q k ) ,θ k ⊂ θ k − 1 F θ k − log 1 /δ − 1 X j = k +1 X θ j ∈ Θ = j ( Q k ) ,θ j ⊂ θ k − 1 F θ j , (4.7) with θ − 1 := S 1 , log 1 /δ − 1 X j =log 1 /δ = log 1 /δ − 1 X j =log 1 /δ +1 := 0 , Θ log 1 /δ := ∅ as conv en tion. In particular, when k = 0, it b ecomes the main term in Lemma 4.2; when k = log 1 /δ , it b ecomes F θ k − 1 for R − 1 / 2 -caps θ k − 1 . W e shall need the following relation b etw een F Q k ,θ k − 1 and F Q k +1 ,θ k . PINNED DIST ANCE SET AND MIZOHA T A-T AKEUCHI-TYPE ESTIMA TES 17 Lemma 4.3. Supp ose Q k ⊂ Q k +1 , Q k ∈ D r k /R , Q k +1 ∈ D r k +1 /R , then F Q k ,θ k − 1 ( y ) = X θ k / ∈ Θ k ( Q k ) , θ k ⊂ θ k − 1 F Q k +1 ,θ k ( y ) , ∀ y ∈ R 2 . Pr o of of L emma 4.3. Notice that F Q k ,θ k − 1 = X θ k ⊂ θ k − 1 F θ k − X θ k ∈ Θ k ( Q k ) , θ k ⊂ θ k − 1 F θ k − X θ k ⊂ θ k − 1 log 1 /δ − 1 X j = k +1 X θ j ∈ Θ = j ( Q k ) , θ j ⊂ θ k F θ j = X θ k / ∈ Θ k ( Q k ) , θ k ⊂ θ k − 1 F θ k − X θ k ⊂ θ k − 1 log 1 /δ − 1 X j = k +1 X θ j ∈ Θ = j ( Q k ) ,θ j ⊂ θ k F θ j . (4.8) By the definition of Θ = j in (4.5), the conditions θ j ∈ Θ = j ( Q k ) , θ j ⊂ θ k , j ⩾ k + 1 imply θ k / ∈ Θ k ( Q k ). Therefore (4.8) equals X θ k / ∈ Θ k ( Q k ) , θ k ⊂ θ k − 1   F θ k − log 1 /δ − 1 X j = k +1 X θ j ∈ Θ = j ( Q k ) ,θ j ⊂ θ k F θ j   . (4.9) As the sum is taken o v er θ k / ∈ Θ k ( Q k ), no θ k +1 ⊂ θ k in tersect caps in Θ k ( Q k ). So (4.9) can b e written as X θ k / ∈ Θ k ( Q k ) , θ k ⊂ θ k − 1 ( F θ k − X θ k +1 ∈ Θ k +1 ( Q k ) , θ k +1 ⊂ θ k F θ k − log 1 /δ − 1 X j = k +2 X θ j ∈ Θ = j ( Q k ) ,θ j ⊂ θ k F θ j ) . Since Θ j ( Q k ) = Θ j ( Q k +1 ) when Q k ⊂ Q k +1 and j ⩾ k + 1 (recall (4.4)), it coincides with X θ k / ∈ Θ k ( Q k ) , θ k ⊂ θ k − 1   F θ k − X θ k +1 ∈ Θ k +1 ( Q k +1 ) , θ k +1 ⊂ θ k F θ k − log 1 /δ − 1 X j = k +2 X θ j ∈ Θ = j ( Q k +1 ) ,θ j ⊂ θ k F θ j   = X θ k / ∈ Θ k ( Q k ) , θ k ⊂ θ k − 1 F Q k +1 ,θ k , as desired. □ No w, by Lemma 4.2 and notation (4.7), the in tegral (4.3) equals, up to a negligible error, X Q 0 ∈D r 0 /R ν ( Q 0 ) Z | F Q 0 ,θ − 1 ( y ) | 2 dν Q 0 ( y ) , (4.10) whic h is, b y the standard lo calization argumen t, ≲ R C δ X Q 0 ∈D r 0 /R ν ( R 2 δ 2 Q 0 ) Z | F Q 0 ,θ − 1 ( y ) | 2 dm R 2 δ 2 Q 0 ( y ) . In this pap er, the constant C > 0 in the exp onent of R may v ary from line to line but is indep endent in δ and R . Here is our k ey lemma on the scale reduction. 18 BOCHEN LIU Lemma 4.4. F or 0 ⩽ k ⩽ log 1 /δ − 1 , e ach Q k +1 ∈ D r k +1 /R , and e ach r − 1 k − 1 -c ap θ k − 1 , X Q k ⊂ Q k +1 ν ( R 2( k +1) δ 2 Q k ) Z | F Q k ,θ k − 1 ( y ) | 2 dm R 2( k +1) δ 2 Q k ( y ) ≲ R C δ ( r k /r k +1 ) min { t, 1 }− 1 X θ k ⊂ θ k − 1 ν ( R 2( k +2) δ 2 Q k +1 ) Z | F Q k +1 ,θ k ( y ) | 2 dm R 2( k +2) δ 2 Q k +1 ( y ) . Pr o of of L emma 4.4. The idea is the same as the heuristic pro of of Theorem 1.2 in Section 4.1, and the dep endence on Q k ∈ D r k /R will b e reduced to Q k +1 ∈ D r k +1 /R in a natural w ay . W e shall giv e a detailed pro of. I hop e that the heuristic argument ab ov e helps for a b etter understanding of this idea. By Lemma 4.3, for eac h Q k ∈ D r k /R , Z | F Q k ,θ k − 1 ( y ) | 2 dm R 2( k +1) δ 2 Q k ( y ) = Z | X θ k / ∈ Θ k ( Q k ) , θ k ⊂ θ k − 1 F Q k +1 ,θ k ( y ) | 2 dm R 2( k +1) δ 2 Q k ( y ) . As the F ourier supports of { F Q k +1 ,θ k } lie in r − 1 k -caps of RS 1 with bounded ov erlapping, they are contained in R r − 1 k -balls centered in RS 1 of b ounded ov erlapping. So b y the lo cal L 2 -orthogonalit y , for each r k /R -cub e Q ′ ⊂ R 2( k +1) δ 2 Q k , Z | X θ k / ∈ Θ k ( Q k ) , θ k ⊂ θ k − 1 F Q k +1 ,θ k | 2 dm Q ′ ≲ R C δ X θ k / ∈ Θ k ( Q k ) , θ k ⊂ θ k − 1 Z | F Q k +1 ,θ k | 2 dm R δ 2 Q ′ , up to an error term RapDec( R ) ∥ F ∥ 2 L 2 . Since there are ≲ R C δ 2 man y suc h Q ′ , it follows that, up to a negligible error, Z | F Q k ,θ k − 1 | 2 dm R 2( k +1) δ 2 Q k = Z | X θ k / ∈ Θ k ( Q k ) , θ k ⊂ θ k − 1 F Q k +1 ,θ k | 2 dm R 2( k +1) δ 2 Q k ≲ R C δ X θ k / ∈ Θ k ( Q k ) , θ k ⊂ θ k − 1 Z | F Q k +1 ,θ k | 2 dm R (2 k +3) δ 2 Q k . (4.11) By taking the sum of (4.11) o ver Q k ⊂ Q k +1 with w eigh t ν ( R 2( k +1) δ 2 Q k ), one can see that the left hand side in Lemma 4.4 satisfies X Q k ⊂ Q k +1 ν ( R 2( k +1) δ 2 Q k ) Z | F Q k ,θ k − 1 | 2 dm R 2( k +1) δ 2 Q k ≲ R C δ X Q k ⊂ Q k +1 X θ k / ∈ Θ k ( Q k ) , θ k ⊂ θ k − 1 ν ( R 2( k +1) δ 2 Q k ) Z | F Q k +1 ,θ k | 2 dm R (2 k +3) δ 2 Q k . (4.12) No w w e fix θ k ⊂ θ k − 1 and count Q k . As θ k / ∈ Θ k ( Q k ), by the definition of Θ k ( Q k ) in (3.9) and (3.8), Q k is counted only if the r k /R × r k +1 /R tub e T r k /R × r k +1 /R cen tered at the center of Q k of direction θ k is “go o d”, meaning ν R 2( k +2) δ 2 Q k +1 ( R δ T r k /R × r k +1 /R ) < R 10 δ · ( r k /r k +1 ) min { t, 1 } . (4.13) PINNED DIST ANCE SET AND MIZOHA T A-T AKEUCHI-TYPE ESTIMA TES 19 So, if w e cov er Q k +1 b y r k /R × r k +1 /R tub es T parallel to θ k of b ounded o verlapping, then (4.12) is ≲ R C δ X θ k ⊂ θ k − 1 X T ∥ θ k with (4.13) r k /R × r k +1 /R -tub es X Q k ∩ T  = ∅ ν ( R 2( k +1) δ 2 Q k ) Z | F Q k +1 ,θ k | 2 dm R (2 k +3) δ 2 Q k . (4.14) F or each T in (4.14), co ver R (2 k +3) δ 2 T b y r k /R × r k +1 /R tub es T ′ of the same direction. As the F ourier supp ort of F Q k +1 ,θ k lies in a R/r k × R/r k +1 rectangle containing θ k , it has lo cally constant prop ert y on each T ′ , and therefore X Q k ∩ T  = ∅ ν ( R 2( k +1) δ 2 Q k ) Z T ′ | F Q k +1 ,θ k | 2 dm R (2 k +3) δ 2 Q k ≲ R C δ · ν ( R 2( k +2) δ 2 T ) · ∥ F Q k +1 ,θ k ∥ 2 L ∞ ( T ′ ) = R C δ · ν R 2( k +2) δ 2 Q k +1 ( R 2( k +2) δ 2 T ) · ν ( R 2( k +2) δ 2 Q k +1 ) · ∥ F Q k +1 ,θ k ∥ 2 L ∞ ( T ′ ) ≲ R C δ · ν R 2( k +2) δ 2 Q k +1 ( R δ T ) · ν ( R 2( k +2) δ 2 Q k +1 ) · m ( R δ 2 T ′ ) − 1 Z R δ 2 T ′ | F Q k +1 ,θ k | 2 dm, whic h is, b ecause T satisfies (4.13) and m ( R δ 2 T ′ ) ≳ R − C δ · ( r k /r k +1 ) · m ( R 2( k +2) δ 2 Q k +1 ), ≲ R C δ · ( r k /r k +1 ) min { t, 1 }− 1 · ν ( R 2( k +2) δ 2 Q k +1 ) Z R δ 2 T ′ | F Q k +1 ,θ k | 2 dm R 2( k +2) δ 2 Q k +1 . (4.15) Finally w e tak e the sum of (4.15) o ver T ′ ⊂ R (2 k +3) δ 2 T and then the sum ov er T , θ k in (4.14) to obtain the follo wing upp er b ound of the left hand side in Lemma 4.4: R C δ ( r k /r k +1 ) min { t, 1 }− 1 X θ k ⊂ θ k − 1 ν ( R 2( k +2) δ 2 Q k +1 ) X T ∥ θ k with (4.13) r k /R × r k +1 /R -tub es Z R 2( k +2) δ 2 T | F Q k +1 ,θ k | 2 dm R 2( k +2) δ 2 Q k +1 . No w w e hav e finished all the work in the scale r − 1 k , can add those “bad” r k /R × r k +1 /R tub es without (4.13) bac k to mov e to the next scale. More precisely , since the union of r k /R × r k +1 /R tub es T cov ers Q k of b ounded ov erlapping, the ab o ve is ≲ R C δ ( r k /r k +1 ) min { t, 1 }− 1 X θ k ⊂ θ k − 1 ν ( R 2( k +2) δ 2 Q k +1 ) Z | F Q k +1 ,θ k | 2 dm R 2( k +2) δ 2 Q k +1 , th us completes the pro of of Lemma 4.4. □ By (4.3), (4.10), and iteration of Lemma 4.4, we end up with Z | µ g ood,i,Q ( y ) ∗ c σ R ( y ) | 2 dν ( y ) ≲ R C δ log 1 /δ log δ − 1 Y k =0 ( r k /r k +1 ) min { t, 1 }− 1 X θ : R − 1 / 2 -caps Z B R δ | F θ | 2 dm ≲ R C δ log 1 /δ R 1 − min { t, 1 } X θ : R − 1 / 2 -caps Z | F θ | 2 dm. 20 BOCHEN LIU The rest is the standard “Plancherel & Cauc hy-Sc h warz” argumen t: X θ : R − 1 / 2 -caps Z | F θ | 2 dm = X θ : R − 1 / 2 -caps Z | (( ψ i,θ b µ dσ R ) ∗ ψ ) ∨ | 2 dm = X θ : R − 1 / 2 -caps Z | ( ψ i,θ b µ dσ R ) ∗ ψ | 2 dm ≲ R − 1 Z S 1 X θ : R − 1 / 2 -caps | ( ψ i,θ b µ )( Rσ ) | 2  Z ψ ( ξ − Rσ ) dξ  dσ ≲ R − 1 Z S 1 | b µ ( Rσ ) | 2 dσ. Hence, Z | µ g ood,i,Q ( y ) ∗ c σ R ( y ) | 2 dν ( y ) ≲ R C δ log 1 /δ R − min { t, 1 } Z S 1 | b µ ( Rσ ) | 2 dσ, and, after integrating o ver R ≈ R i , it follows that Z Z r ≈ R i | µ g ood,y ∗ b σ r ( y ) | 2 r dr dν ( y ) ≲ δ R C δ log 1 /δ i R − min { t, 1 } i Z | ξ |≈ R i | b µ ( ξ ) | 2 dξ ≲ δ R C δ log 1 /δ i R 2 − s − min { t, 1 } i I s ( µ ) , summable in i if s > 1, s + t > 2, and δ > 0 is small enough in terms of s, t . This completes the pro of of Prop osition 4.1. 5. Pr oof of the main Theorem No w w e can pro ve Theorem 1.1. By the discussion in Section 2, there exist a proba- bilit y measure µ on E of finite s -energy I s ( µ ) for some s > 1, and a probability measure ν on F that is F rostman of dimension t > 2 − s with dim M supp ν < t + δ 2 . Then, by Prop osition 3.2, Prop osition 4.1, and 1 = Z Z ∆ y ( E ) d y ∗ ( µ ) ! dν ( y ) ⩽ ∥ d y ∗ ( µ g ood,y ) − d y ∗ ( µ )) ∥ L 1 +( sup y ∈ supp ν | ∆ y ( E ) | 1 2 ) ∥ d y ∗ ( µ g ood,y ) ∥ L 2 , one can conclude that | ∆ y ( E ) | > 0 , for some y ∈ F , that completes the pro of of Theorem 1.1. References [1] Bal´ azs B´ ar´ any . On some non-linear pro jections of self-similar sets in R 3 . F und. Math. , 237(1):83– 100, 2017. doi:10.4064/fm90- 4- 2016 . [2] Jean B ourgain. Hausdorff dimension and distance sets. Isr ael J. Math. , 87(1-3):193–201, 1994. URL: http://dx.doi.org/10.1007/BF02772994 , doi:10.1007/BF02772994 . [3] Jean Bourgain. On the Erd˝ os-Volkmann and Katz-Tao ring conjectures. Ge om. F unct. Anal . , 13(2):334–365, 2003. URL: http://dx.doi.org/10.1007/s000390300008 , doi:10.1007/ s000390300008 . [4] Hannah Cairo. A counterexample to the mizohata-takeuc hi conjecture. 2025. URL: https:// arxiv.org/abs/2502.06137 , arXiv:2502.06137 . [5] Hannah Cairo and Ruixiang Zhang. Po wer loss for the Mizohata-Takeuc hi conjecture on C k con vex h yp ersurfaces. 2025. URL: , . PINNED DIST ANCE SET AND MIZOHA T A-T AKEUCHI-TYPE ESTIMA TES 21 [6] Anthon y Carb ery , Timo S. H¨ anninen, and Stef´ an Ingi V aldimarsson. Disen tanglement, m ultilinear dualit y and factorisation for nonp ositive op erators. Anal. PDE , 16(2):511–543, 2023. doi:10. 2140/apde.2023.16.511 . [7] Anthon y Carb ery , Marina Iliop oulou, and Hong W ang. Some sharp inequalities of Mizohata- Tak euchi-t yp e. R ev. Mat. Ib er o am. , 40(4):1387–1418, 2024. doi:10.4171/rmi/1463 . [8] Anthon y Carb ery , Zane Kun Li, Yixuan P ang, and Po-Lam Y ung. A weigh ted form ulation of refined decoupling and inequalities of mizohata-takeuc hi-type for the moment curve, 2025. URL: https://arxiv.org/abs/2510.04345 , arXiv:2510.04345 . [9] Ciprian Demeter. F ourier r estriction, de c oupling, and applic ations , v olume 184 of Cambridge Studies in A dvanc e d Mathematics . Cambridge Universit y Press, Cam bridge, 2020. doi:10.1017/ 9781108584401 . [10] Xiumin Du, Larry Guth, Y umeng Ou, Hong W ang, Bobb y Wilson, and Ruixiang Zhang. W eighted restriction estimates and application to Falconer distance set problem. Amer. J. Math. , 143(1):175– 211, 2021. doi:10.1353/ajm.2021.0005 . [11] Xiumin Du, Alex Iosevic h, Y umeng Ou, Hong W ang, and Ruixiang Zhang. An impro ved result for Falconer’s distance set problem in even dimensions. Math. Ann. , 380(3-4):1215–1231, 2021. doi:10.1007/s00208- 021- 02170- 1 . [12] Xiumin Du, Y umeng Ou, Kevin Ren, and Ruixiang Zhang. New impro vemen t to falconer distance set problem in higher dimensions. 2023. URL: , arXiv:2309.04103 . [13] Xiumin Du, Y umeng Ou, Kevin Ren, and Ruixiang Zhang. W eighted refined decoupling estimates and application to falconer distance set problem. 2023. URL: 04501 , arXiv:2309.04501 . [14] Xiumin Du and Ruixiang Zhang. Sharp L 2 estimates of the Sc hr¨ odinger maximal function in higher dimensions. Ann. of Math. (2) , 189(3):837–861, 2019. URL: https://doi- org.easyaccess1.lib. cuhk.edu.hk/10.4007/annals.2019.189.3.4 , doi:10.4007/annals.2019.189.3.4 . [15] M. Burak Erdo˘ gan. A bilinear Fourier extension theorem and applications to the distance set problem. Int. Math. R es. Not. , (23):1411–1425, 2005. URL: http://dx.doi.org/10.1155/IMRN. 2005.1411 , doi:10.1155/IMRN.2005.1411 . [16] K. J. F alconer. On the Hausdorff dimensions of distance sets. Mathematika , 32(2):206–212, 1985. URL: http://dx.doi.org/10.1112/S0025579300010998 , doi:10.1112/S0025579300010998 . [17] K. J. F alconer. The ge ometry of fr actal sets , volume 85 of Cambridge T r acts in Mathematics . Cam bridge Universit y Press, Cambridge, 1986. [18] Andrew F erguson, Jonathan M. F raser, and T uomas Sahlsten. Scaling scenery of ( × m, × n ) inv ari- an t measures. A dv. Math. , 268:564–602, 2015. doi:10.1016/j.aim.2014.09.019 . [19] Jacob B. Fiedler and D. M. Stull. Dimension of pinned distance sets for semi-regular sets. 2023. URL: , . [20] Jacob B. Fiedler and D. M. Stull. Pinned distances of planar sets with low dimension. 2024. URL: https://arxiv.org/abs/2408.00889 , arXiv:2408.00889 . [21] Jonathan M. F raser. Distance sets, orthogonal pro jections and passing to weak tangents. Isr ael J. Math. , 226(2):851–875, 2018. doi:10.1007/s11856- 018- 1715- z . [22] Jonathan M. F raser. A nonlinear pro jection theorem for Assouad dimension and applications. J. L ond. Math. So c. (2) , 107(2):777–797, 2023. doi:10.1112/jlms.12697 . [23] Jonathan M. F raser and Mark Pollicott. Micromeasure distributions and applications for con- formally generated fractals. Math. Pr o c. Cambridge Philos. So c. , 159(3):547–566, 2015. doi: 10.1017/S0305004115000523 . [24] Larry Guth. An enem y scenario in restriction theory . Joint talk for AIM Research Comm unity ‘F ourier restriction conjecture and related problems’ and HAPPY netw ork., 2022. URL: https: //www.youtube.com/watch?v=x- DET83UjFg . [25] Larry Guth, Alex Iosevic h, Y umeng Ou, and Hong W ang. On Falconer’s distance set problem in the plane. Invent. Math. , 219(3):779–830, 2020. doi:10.1007/s00222- 019- 00917- x . [26] Alex Iosevich and Bo chen Liu. F alconer distance problem, additive energy and cartesian pro ducts. Ann. A c ad. Sci. F enn. Math. , 41(2):579–585, 2016. doi:doi:10.5186/aasfm.2016.4135 . [27] Nets Hawk Katz and T erence T ao. Some connections betw een Falconer’s distance set conjecture and sets of Furstenburg type. New Y ork J. Math. , 7:149–187, 2001. URL: http://nyjm.albany. edu:8000/j/2001/7_149.html . 22 BOCHEN LIU [28] T am´ as Keleti and P ablo Shmerkin. New b ounds on the dimensions of planar distance sets. Ge om. F unct. Anal. , 29(6):1886–1948, 2019. doi:10.1007/s00039- 019- 00500- 9 . [29] Bo chen Liu. An L 2 -iden tity and pinned distance problem. Ge om. F unct. A nal. , 29(1):283–294, 2019. URL: https://doi- org.easyaccess1.lib.cuhk.edu.hk/10.1007/s00039- 019- 00482- 8 , doi:10.1007/s00039- 019- 00482- 8 . [30] Bo chen Liu. Hausdorff dimension of pinned distance sets and the L 2 -metho d. Pr o c. A mer. Math. So c. , 148(1):333–341, 2020. doi:10.1090/proc/14740 . [31] Bo chen Liu. Mixed-norm of orthogonal pro jections and analytic interpolation on dimensions of measures. R ev. Mat. Ib er o am. , 40(3):827–858, 2024. doi:10.4171/rmi/1472 . [32] Pertti Mattila. Spherical a v erages of Fourier transforms of measures with finite energy; dimension of in tersections and distance sets. Mathematika , 34(2):207–228, 1987. URL: http://dx.doi.org/ 10.1112/S0025579300013462 , doi:10.1112/S0025579300013462 . [33] Pertti Mattila. Ge ometry of Sets and Me asur es in Euclide an Sp ac es: F r actals and R e cti- fiability , volume 44 of Cambridge Studies in A dvanc e d Mathematics . Cam bridge Univ ersity Press, Cambridge, 1995. URL: http://dx.doi.org/10.1017/CBO9780511623813 , doi:10.1017/ CBO9780511623813 . [34] Siddharth Mulherk ar. Random constructions for sharp estimates of mizohata-takeuc hi type, 2025. URL: , . [35] T uomas Orponen. On the distance sets of self-similar sets. Nonline arity , 25(6):1919–1929, 2012. doi:10.1088/0951- 7715/25/6/1919 . [36] T uomas Orp onen. On the distance sets of Ahlfors-David regular sets. A dv. Math. , 307:1029–1045, 2017. URL: http://dx.doi.org/10.1016/j.aim.2016.11.035 , doi:10.1016/j.aim.2016.11. 035 . [37] T uomas Orponen. On the dimension and smo othness of radial pro jections. A nal. PDE , 12(5):1273– 1294, 2019. URL: https://doi- org.easyaccess2.lib.cuhk.edu.hk/10.2140/apde.2019.12. 1273 , doi:10.2140/apde.2019.12.1273 . [38] Y uv al P eres and Wilhelm Schlag. Smo othness of pro jections, Bernoulli con volutions, and the dimension of exceptions. Duke Math. J. , 102(2):193–251, 2000. URL: http://dx.doi.org/10. 1215/S0012- 7094- 00- 10222- 0 , doi:10.1215/S0012- 7094- 00- 10222- 0 . [39] Pablo Shmerkin. A nonlinear version of b ourgain’s pro jection theorem. J. Eur. Math. So c. (2022), DOI:10.4171/JEMS/1283 . [40] Pablo Shmerkin. On the Hausdorff dimension of pinned distance sets. Isr ael J. Math. , 230(2):949–972, 2019. URL: https://doi- org.easyaccess1.lib.cuhk.edu.hk/10.1007/ s11856- 019- 1847- 9 , doi:10.1007/s11856- 019- 1847- 9 . [41] Pablo Shmerkin. Impro ved bounds for the dimensions of planar distance sets. J. F r actal Ge om. , 8(1):27–51, 2021. doi:10.4171/jfg/97 . [42] Pablo Shmerkin and Hong W ang. On the distance sets spanned by sets of dimension d/ 2 in R d . Ge om. F unct. A nal. , 35(1):283–358, 2025. doi:10.1007/s00039- 024- 00696- 5 . [43] Elias M. Stein. Harmonic analysis: r e al-variable metho ds, ortho gonality, and oscil latory inte gr als , v olume 43 of Princ eton Mathematic al Series . Princeton Universit y Press, Princeton, NJ, 1993. With the assistance of Timoth y S. Murph y , Monographs in Harmonic Analysis, I I I. [44] D. M. Stull. Pinned distance sets using effective dimension. 2022. URL: 2207.12501 , arXiv:2207.12501 . [45] Zijian W ang and Jiqiang Zheng. An improv emen t of the pinned distance set problem in even dimensions. Col lo q. Math. , 170(2):171–191, 2022. doi:10.4064/cm8632- 10- 2021 . [46] Thomas W olff. Deca y of circular means of Fourier transforms of measures. Internat. Math. R es. Notic es , (10):547–567, 1999. URL: http://dx.doi.org/10.1155/S1073792899000288 , doi:10. 1155/S1073792899000288 . Email addr ess : Bochen.Liu1989@gmail.com Dep ar tment of Ma thema tics & Interna tional Center for Ma thema tics, Southern University of Science and Technology, Shenzhen 518055, China