Furstenberg-type estimates under mild non-concentration assumptions

FURSTENBERG-TYPE ESTIMA TES UNDER MILD NON-CONCENTRA TION ASSUMPTIONS TUOMAS ORPONEN AND P ABLO SHMERKIN A B S T R A C T . W e prove sharp δ -discretised versions of some variants of the Fursten- berg set pr oblem under weaker or differ ent non-concentration assumptions com- pared to pr evious works. C O N T E N T S 1. Introduction and main r esults 1 2. Preliminaries and notation 5 2.1. Dyadic scales, tubes, and nice configurations 5 2.2. Uniform sets and branching functions 6 2.3. Lipschitz decomposition lemma 6 2.4. Multiscale decomposition of nice configurations 7 3. Proof of Theor em 1.2 7 4. Proof of Theor ems 1.4 and 1.5 12 5. Sharpness of main results 15 5.1. Sharpness of Theorem 1.2 and Cor ollary 1.3. 15 5.2. Sharpness of Theorem 1.4 16 5.3. Sharpness of Theorem 1.5 16 Appendix A. Ratios of arithmetic progr essions 19 References 21 1. I N T R O D U C T I O N A N D M A I N R E S U LT S The study of projections and mor e general discretised incidence pr oblems has seen many significant developments in recent years. Notably , the Furstenberg set conjectur e was recently resolved by Ren and W ang [ 10 ], building up on our previous work [ 7 ]. W e next state the δ -discretised version of this estimate. The Date : 20th March 2026. 2010 Mathematics Subject Classification. 28A80 (primary) 28A78 (secondary). Key words and phrases. Pr ojections, Furstenberg sets. T .O. is supported by the European Research Council (ERC) under the European Union’s Ho- rizon Europe resear ch and innovation pr ogramme (grant agreement No 101087499), and by the Research Council of Finland via the pr oject Approximate incidence geometry , grant no. 355453. P .S. is supported by an NSERC Discovery Grant. 1 2 TUOMAS ORPONEN AND P ABLO SHMERKIN original conjecture, due to W olff [ 16 ], was stated in terms of Hausdorff dimen- sion, but the δ -discr etised formulation is more powerful in applications, such as the ones in this paper . W e refer to Section 2 for the definitions of the discretised objects (for example Frostman sets and configurations) appearing in the state- ment. Here we just informally mention that a p δ , s, C , M q -configuration p P , T q is a pair of δ -squar es P and δ -tubes T such that each squar e p P P intersects all tubes in an s -dimensional sub-family T p p q Ă T of cardinality | T p p q| “ M . Theorem 1.1 (Furstenberg set estimate) . Let s P p 0 , 1 s , t P r 0 , 2 s , and let γ ă min ␣ t, s ` t 2 , 1 ( . Then, there exist ϵ “ ϵ p s, t, γ q ą 0 and δ 0 “ δ 0 p s, t, γ q ą 0 such that the following holds for all δ P p 0 , δ 0 s . Let p P , T q be a p δ, s, δ ´ ϵ , M q -nice configuration, where P is a Frostman p δ , t, δ ´ ϵ q -set. Then, | T | ě M δ ´ γ . Moreover , δ 0 , ϵ can be chosen uniform over all p s, t, γ q in a fixed compact subset of p 0 , 1 s ˆ r 0 , 2 s ˆ p 0 , min t t, s ` t 2 , 1 uqq . Theorem 1.1 is [ 10 , Theorem 4.1]. Many variants of Theorem 1.1 have been achieved, with important applications to the restriction conjecture and discret- ised sum-product problems, among others. See [ 2 , 3 , 14 , 15 ]. Each of these vari- ants involves some relaxation of the non-concentration assumptions on the tubes T (often at the cost of some additional assumptions on the set P ). The main goal of this paper is to obtain versions of Theorem 1.1 which are valid under weaker non-concentration assumptions on the set P , but keeping a strong non-concentration assumption on the tube family T . The first main result is Theorem 1.2 below . The point of this result is to obtain an improvement over the "trivial" bound | T | Ç M | P | 1 { 2 (see for example [ 9 , Pr oposition 2.9] for a precise statement) under the hypothesis that P is not concentrated in a single square of (minimal) side-length δ | P | 1 { 2 . Theorem 1.2. Fix s P p 0 , 1 s , t P r 0 , 2 s , u P p 0 , min t t, 2 ´ t us , and 0 ă ζ ă t 2 ` s ¨ u 2 . Then, ther e exist ϵ “ ϵ p ζ , s, t, u q ą 0 and δ 0 “ δ 0 p ζ , s, t, u q ą 0 such that the following holds for all δ P 2 ´ N X p 0 , δ 0 s . Let p P , T q be a p δ, s, δ ´ ϵ , M q -nice configuration. Assume that | P | “ δ ´ t and | P X Q | ď δ u | P | , Q P D δ | P | 1 { 2 . (1.1) Then | T | ě M ¨ δ ´ ζ . (1.2) FURSTENBERG TYPE ESTIMA TES 3 It is well known that discretised Furstenberg set estimates imply discretised projection estimates. In short, for an s -Frostman set Θ Ă S 1 of dir ections, one con- siders for each p P P the tubes T p p q through p in dir ections θ P Θ . Then a Fursten- berg set estimate applied to the p δ, s, C , | Θ |q -nice configuration p P , Ť p T p p qq im- plies a projection estimate. W ith this ar gument, Theorem 1.2 implies the follow- ing projection estimate under minimal non-concentration assumptions: Corollary 1.3. Fix s, t, u , and ζ as in Theor em 1.2 . Then, there exist ϵ “ ϵ p ζ , s, t, u q ą 0 and δ 0 “ δ 0 p ζ , s, t, u q ą 0 such that the following holds for all δ P 2 ´ N X p 0 , δ 0 s . Assume that P Ă D δ has | P | δ “ δ ´ t , and satisfies ( 1.1 ) . Let Θ Ă S 1 be a non-empty Frostman p δ , s, δ ´ ϵ q -set. Then, there exists θ P Θ such that | π θ p P 1 q| δ ě δ ´ ζ , P 1 Ă P , | P 1 | ě δ ϵ | P | . (1.3) Here π θ p x q : “ x ¨ θ for x P R 2 . The single-scale non-concentration assumption ( 1.1 ) first appeared in [ 11 ], where the second author proved a non-quantitative version of Corollary 1.3 . W e note that in Theorems 1.1 and 1.2 , and in Corollary 1.3 , the cases t P t 0 , 2 u are easily checked to be trivial, but it is convenient to include them in the statement. Theorem 1.2 and Cor ollary 1.3 are sharp; this will be shown in § 5.1 . W e then move to the second main theorem of the paper , where the "minimal" non-concentration ( 1.1 ) on P is upgraded to a Katz-T ao non-concentration hypo- thesis (see Definition 2.1 ). Theorem 1.4. Let s P p 0 , 1 s , t P r s, 2 ´ s s , and η ą 0 . Then ther e exist ϵ “ ϵ p s, t, η q ą 0 and δ 0 “ δ 0 p s, t, η q ą 0 such that the following holds for all δ P p 0 , δ 0 s . Let P Ă D δ be a Katz-T ao p δ , t, δ ´ ϵ q -set, and let p P , T q be a p δ , s, δ ´ ϵ , M q -nice configuration. Then, | T | ě δ η M | P | p s ` t q{p 2 t q . Theorem 1.4 is also sharp; see § 5.2 . The sharpness examples have the form P “ A ˆ A , where A Ă δ Z X r 0 , 1 s is a Katz-T ao p δ, t { 2 q -set. Is Theorem 1.4 also sharp for pr oduct sets P “ A ˆ B , wher e | A | ‰ | B | ? Not always: even the trivial bound | π θ p P q| δ Á | A | may sometimes beat the bound p| A || B |q p 1 ` s q{ 2 suggested by applying Theorem 1.4 directly to P “ A ˆ B . Motivated by this observation, we present a variant of Theorem 1.4 for (generally) asymmetric product sets. This comes at negligible additional cost, since the proof of Theorem 1.5 is almost the same as the proof of Theor em 1.4 . Theorem 1.5. Let s P p 0 , 1 s , let α , β P p 0 , 1 s with α ` β P r s, 2 ´ s s , and let η ą 0 . Then there exist ϵ “ ϵ p s, α, β , η q ą 0 and δ 0 “ δ 0 p s, α, β , η q ą 0 such that the following holds for all δ P p 0 , δ 0 s . ‚ Let A Ă δ Z X r 0 , 1 s be a Katz-T ao p δ, α, δ ´ ϵ q -set. ‚ Let B Ă δ Z X r 0 , 1 s be a Katz-T ao p δ, β , δ ´ ϵ q -set. 4 TUOMAS ORPONEN AND P ABLO SHMERKIN Assume that p A ˆ B , T q is a p δ, s, δ ´ ϵ , M q -nice configuration, and | A | β ě | B | α . Then, | T | ě δ η M # | A || B | p β ` s ´ α q{p 2 β q , α ď s, | A | p α ` s q{p 2 α q | B | 1 { 2 , α ě s. (1.4) Remark 1.6. W e only examined the sharpness of Theorem 1.5 in the case α “ β . Then, for | A | ě | B | , ( 1.4 ) and 2 α P r s, 2 ´ s s , the bound ( 1.4 ) simplifies to | T | ě δ η M # | A || B | s {p 2 α q , α ď s, | A | p α ` s q{p 2 α q | B | 1 { 2 , α ě s. (1.5) It turns out that the first bound is always sharp, and the second bound is sharp whenever | A | α ´ s ď | B | α . In fact, for | A | α ´ s “ | B | α the second bound simplifies to | T | ě δ η M | A | . In the range | B | α ă | A | α ´ s , the "trivial bound" | T | Ç M | A | is the sharp bound (and is stronger than the one given by Theor em 1.5 ). W e will sketch sharpness examples to ( 1.5 ) in Section 5.3 . Theorems 1.4 – 1.5 should be contrasted with the recent results of H. W ang and S. W u [ 14 , 15 ]. The differ ence is that the non-concentration condition of Theor em 1.4 for the tube families T p p q incident to the squar es p P P is significantly str onger than in [ 14 , 15 ]. On the other hand the conclusions are also stronger in cases where M or | P | are small. Let us give one concrete comparison: in the case t “ 1 the main theorem of [ 14 ] applied to the setting of Theorem 1.4 yields the lower bound | T | Ç M 3 { 2 δ 1 { 2 | P | . The lower bound is smaller (weaker) than the one in Theorem 1.4 whenever M | P | 1 ´ s ă δ ´ 1 . In fact, even the "trivial" bound | T | Ç M | P | 1 { 2 beats [ 14 ] when M | P | ă δ ´ 1 . The proofs of all our theorems follow the “combinatorial scale decomposi- tion” appr oach. Applying Lipschitz function combinatorial decompositions fr om [ 11 , 13 ], we ar e able to locate scale blocks on which Theorem 1.1 can be applied. The details of the multiscale decomposition are new . The differ ent scale blocks are combined via a multi-scale incidence estimate from [ 7 ]. This background is recalled in Section 2 , and the proofs of the main theorems are given in Sections 3 and 4 . The sharpness examples are discussed in Section 5 . Remark 1.7. W e comment on the history of this project. Initially , a non-sharp version of Theor em 1.2 (with su { 4 instead of su { 2 ) was obtained in the first arXiv version of our article [ 8 ] in 2023. This estimate was based on the “sticky” Fursten- berg estimate obtained in [ 8 ]. A few months later , K. Ren and H. W ang proved the sharp Furstenberg set estimate in Theor em 1.1 . W e realised that by applying their result we could obtain the sharp version of Theor em 1.2 , stated above. Since the sharp version Theorem 1.2 relies on the sharp Furstenberg set estimate that appeared after our preprint, and following a refer ee suggestion, we decided to split of f Theor em 1.2 fr om [ 8 ] into the present article. In addition to strengthening the result, we have also added and corrected details in the proofs with r espect to the arXiv version of [ 8 ]. Theorems 1.4 and 1.5 appear her e for the first time. FURSTENBERG TYPE ESTIMA TES 5 2. P R E L I M I N A R I E S A N D N O TA T I O N 2.1. Dyadic scales, tubes, and nice configurations. W e fix dyadic scales δ, ∆ P 2 ´ N and follow the conventions in [ 8 , Section 2 and Appendix A]. For d P N and A Ă R d , write D δ p A q : “ t p P δ Z d ` r 0 , δ q d : p X A ‰ Hu for the δ -dyadic squar es hitting A . W e abbreviate D δ : “ D δ pr´ 1 , 1 q d q . W e denote | A | δ “ | D δ p A q| . W e extend this to non-dyadic δ by setting | A | δ : “ | A | δ 1 where δ 1 “ min t 2 ´ n : 2 ´ n ď δ u . Up to a multiplicative constant, | A | δ equals the δ -covering number of A . Definition 2.1 (Frostman and Katz-T ao p δ, s, C q -sets) . Let δ P 2 ´ N , s ě 0 and C ą 0 . A set P Ă D δ is called a Frostman p δ , s, C q -set if | P X Q | ď C r s | P | , Q P D r , r P 2 ´ N X r δ, 1 s . Similarly , a set P Ă D δ is called a Katz-T ao p δ, s, C q -set if | P X Q | ď C p r { δ q s , Q P D r , r P 2 ´ N X r δ, 1 s . If the constant " C " is absolute, we sometimes simply write "Frostman p δ, s q -set" or "Katz-T ao p δ, s q -set". W e then proceed to define dyadic δ -tubes . For p a, b q P R 2 , let D p a, b q : “ tp x, y q P R 2 : y “ ax ` b u . This is the “point-line duality” map. If p “ r a, a ` δ q ˆ r b, b ` δ q P D δ pr´ 1 , 1 q ˆ R q , define the associated dyadic δ -tube by D p p q : “ ď p a 1 ,b 1 qP p D p a 1 , b 1 q . W e set T δ : “ t D p p q : p P D δ pr´ 1 , 1 q ˆ R qu . For T “ D pr a, a ` δ q ˆ r b, b ` δ qq P T δ , define its slope by σ p T q : “ a . W e identify subsets P Ă D δ with the union of the squares in P . For a pair p P , T q Ă D δ ˆ T δ and p P P , write T p p q : “ t T P T : T X p ‰ Hu , σ p T p p qq : “ t σ p T q : T P T p p qu . W e say that p P , T q is a p δ , s, C, M q -nice configuration if for every p P P , C ´ 1 M ď | T p p q| ď C M , and σ p T p p qq is a Frostman p δ , s, C q -set in r´ 1 , 1 q . Finally , for p P D ∆ let S p : R 2 Ñ R 2 be the orientation-preserving homothety mapping p onto r 0 , 1 q 2 . For P Ă D δ with δ ď ∆ , S p p P X p q : “ t S p p q q : q P P , q Ă p u Ă D δ { ∆ . W e refer to [ 8 , Section 2] for mor e details on the above definitions. 6 TUOMAS ORPONEN AND P ABLO SHMERKIN 2.2. Uniform sets and branching functions. Assume that δ “ ∆ m with ∆ “ 2 ´ T P 2 ´ N , and let P Ă D δ . W e say that P is t ∆ j u m j “ 1 -uniform if ther e exist numbers N 1 , . . . , N m ě 1 such that for every 1 ď j ď m and every p P D ∆ j ´ 1 p P q , ˇ ˇ D ∆ j p P q X p ˇ ˇ “ N j . In this case, the branching function β : r 0 , m s Ñ r 0 , 2 m s of P is defined by β p j q : “ log | P | ∆ j T “ 1 T j ÿ i “ 1 log N i , 0 ď j ď m, and then linearly interpolating for non-integer values of the parameter . Note that β is 2 -Lipschitz, non-decr easing, piecewise-linear , and satisfies β p 0 q “ 0 and β p m q “ log | P |{ log p 1 { ∆ q . The next standard pigeonholing lemma allows us to pass to a large uniform subset (see [ 8 , Appendix A]). Lemma 2.2. Let ϵ ą 0 . Then there exists T 0 “ T 0 p ϵ q P N such that the following holds. If ∆ “ 2 ´ T with T ě T 0 , δ “ ∆ m , and P Ă D δ , then there exists a t ∆ j u m j “ 1 -uniform set P 1 Ă P such that | P 1 | ě δ ϵ | P | . In particular , one may arrange T ´ 1 log p 2 T q ď ϵ . 2.3. Lipschitz decomposition lemma. Given f : r a, b s Ñ R and numbers ϵ ą 0 , σ , we say that p f , a, b q is p σ, ϵ q -superlinear if f p x q ě f p a q ` σ p x ´ a q ´ ϵ p b ´ a q for all x P r a, b s . The next lemma is [ 8 , Lemma 2.11] (and it follows directly by combining [ 13 , Lemmas 5.21–5.22]). Lemma 2.3. For every d P N and ϵ ą 0 there is τ “ τ p d, ϵ q ą 0 such that the following holds: for any non-decr easing d -Lipschitz function f : r 0 , m s Ñ R with f p 0 q “ 0 there exist sequences 0 “ a 0 ă a 1 ă ¨ ¨ ¨ ă a n “ m, 0 ď t 0 ă t 1 ă ¨ ¨ ¨ ă t n ´ 1 ď d, such that: (i) a j ` 1 ´ a j ě τ m . (ii) p f , a j , a j ` 1 q is p t j , 0 q -superlinear . (iii) ř n ´ 1 j “ 0 p a j ` 1 ´ a j q t j ě f p m q ´ ϵm . Remark 2.4. Note that if F is the piecewise affine function with F p 0 q “ 0 and slope t j on r a j , a j ` 1 s , then (ii) and (iii) imply F p a j q ď f p a j q ď F p a j q ` ϵm, j P t 0 , . . . , n ´ 1 u , and consequently | f p a i q ´ f p a j q ´ p F p a i q ´ F p a j qq| ď ϵm, i, j P t 0 , . . . , n ´ 1 u . FURSTENBERG TYPE ESTIMA TES 7 2.4. Multiscale decomposition of nice configurations. W e now repeat [ 12 , Co- rollary 4.1] (itself based on [ 7 , Proposition 5.1]). In the statement, A Æ δ B stands for A ď C log p 1 { δ q C B for some universal constant C ą 0 , and likewise for A « δ B . Corollary 2.5. Fix N ě 2 and a sequence t ∆ j u N j “ 0 Ă 2 ´ N with 0 ă δ “ ∆ N ă ∆ N ´ 1 ă . . . ă ∆ 1 ă ∆ 0 “ 1 . Let p P 0 , T q Ă D δ ˆ T δ be a p δ, s, C , M q -nice configuration. Then, there exists a set P Ă P 0 such that the following properties hold: (i) | D ∆ j p P q| « δ | D ∆ j p P 0 q| and | P X p | « δ | P 0 X p | , 1 ď j ď N , p P D ∆ j p P q . (ii) For every 0 ď j ď N ´ 1 and p P D ∆ j p P q , there exist numbers C p « δ C and M p ě 1 , and a family of tubes T p Ă T ∆ j ` 1 { ∆ j with the property that p S p p P X p q , T p q is a p ∆ j ` 1 { ∆ j , s, C p , M p q -nice configuration. Furthermore, the families T p can be chosen so that | T | M Ç δ N ´ 1 ź j “ 0 max p j P D ∆ j | T p j | M p j . All the constants implicit in the « δ notation are allowed to depend on N . 3. P R O O F O F T H E O R E M 1 . 2 The idea of the proof of Theorem 1.2 is to first apply to (the branching func- tion of) P a multiscale decomposition provided by Lemma 2.3 . This procedure decomposes the scales between δ and 1 into "blocks" where P looks roughly t j - dimensional for an incr easing sequence of exponents t j P r 0 , 2 s . On each "block" separately , we obtain a lower bound from Theorem 1.1 , and eventually combine the estimates by applying Corollary 2.5 . Remark 3.1. T o facilitate following the next proof, let us note that min t t, s ` t 2 , 1 u “ $ ’ & ’ % t, t ď s, s ` t 2 , t P r s, 2 ´ s s , 1 , t P r 2 ´ s, 2 s . (3.1) Proof of Theor em 1.2 . Fix the parameters s, t, u and 0 ă ζ ă t 2 ` s ¨ u 2 from the state- ment. W rite ω : “ t 2 ` su 2 ´ ζ ą 0 , and choose 0 ă η ă ω 8 . Let ξ “ ξ p s, t, u, ζ , η q ą 0 be another small parameter with ξ ă ω 8 . 8 TUOMAS ORPONEN AND P ABLO SHMERKIN T o avoid endpoint issues, we always apply Theorem 1.1 with a strict margin η below the corresponding endpoint value of γ . Set K η : “ tp s, t 1 , γ 1 q : t 1 P r 0 , 2 s , η ď γ 1 ď min t t 1 , s ` t 1 2 , 1 u ´ η u . All parameter triples used below in applications of Theorem 1.1 (namely for in- dices in I big 1 Y I 2 Y I 3 ) lie in K η , and by the last sentence of Theorem 1.1 we may choose ϵ F “ ϵ F p s, t, u, ζ , η q ą 0 and the small-scale threshold δ 0 uniformly on this compact set. Let τ “ τ p 2 , ξ q ą 0 be the constant fr om Lemma 2.3 , and choose ϵ “ ϵ p s, t, u, ζ , η , ξ q ą 0 so small that ϵ ă min " ω 2 s ` 4 , τ ϵ F 2 * . In particular , sϵ 2 ` 2 η ` 3 ξ 2 ă ω and 2 ϵ τ ă ϵ F . W ithout loss of generality , we may assume that δ ą 0 has the form δ “ ∆ m for some ∆ „ ϵ 1 and m P N , and that P is t ∆ j u m j “ 1 -uniform. This reduction is pos- sible thanks to Lemma 2.2 , which in any case allows us to find a t ∆ j u m j “ 1 -uniform subset P 1 Ă P with | P 1 | ě δ ϵ | P | , provided that ∆ “ 2 ´ T with T ´ 1 log p 2 T q ď ϵ . Let β : r 0 , m s Ñ r 0 , 2 m s be the branching function of P defined in § 2.2 . Apply Lemma 2.3 to β , with parameters d “ 2 and ξ . Let t a j u n j “ 0 , t t j u n ´ 1 j “ 0 be the resulting objects; then n “ O ξ p 1 q by (i). Let F be the piecewise affine function fr om Remark 2.4 . In particular , | β p a i q ´ β p a j q ´ ` F p a i q ´ F p a j q ˘ | ď ξ m, i, j P t 0 , . . . , n ´ 1 u . (3.2) Recall that | P | “ δ ´ t . Then the non-concentration assumption ( 1.1 ) translates into β pp 1 ´ t { 2 q m q ě p u ´ ϵ q m. (3.3) (The ϵm loss accounts for the fact that we had to extract a large uniform subset from the original P .) W e will use the notation Æ δ to hide polylogarithmic losses. W e apply Corollary 2.5 to the sequence ∆ a j , j “ n, n ´ 1 , . . . , 0 . W ith the nota- tion of the corollary , | T | M Ç δ n ´ 1 ź j “ 0 | T p j | M j , where, for brevity , we write M j : “ M p j . Motivated by ( 3.1 ), we split the indices t 0 , . . . , n ´ 1 u into 3 classes: I 1 “ t j : t j P r 0 , s su , I 2 “ t j : t j P p s, 2 ´ s qu , I 3 “ t j : t j P r 2 ´ s, 2 su . Also split I big 1 : “ t j P I 1 : t j ě 2 η u , I small 1 : “ I 1 z I big 1 . FURSTENBERG TYPE ESTIMA TES 9 Recall that the t j are incr easing. Let r 0 , A 1 s “ Y j P I 1 r a j , a j ` 1 s , r A 1 , A 2 s “ Y j P I 2 r a j , a j ` 1 s , r A 2 , m s “ Y j P I 3 r a j , a j ` 1 s , with the convention that A 1 “ 0 if I 1 “ H , A 2 “ A 1 if I 2 “ H , and A 2 “ m if I 3 “ H (it is easy to see that all these cases are consistent). Next, we will estimate each factor | T p j |{ M j individually . The idea is always the same: according to Cor ollary 2.5 the tube family T p j is associated with a p ∆ a j ` 1 ´ a j , s, C j , M j q -nice configuration p P j , T p j q , where P j is a Frostman p ∆ a j ` 1 ´ a j , t j q -set (this follows from part (ii) of Lemma 2.3 and the branching function-to-Frostman dictionary , see e.g. [ 7 , Lemma 8.3]). Now , the range of t j ( t j ď s or t j P p s, 2 ´ s q or t j ě 2 ´ s ) determines the precise form of the lower bound produced by Theorem 1.1 for | T p j |{ M j . W rite ∆ j : “ ∆ a j ` 1 ´ a j . By Corollary 2.5 , we have C j Æ δ δ ´ ϵ . Since a j ` 1 ´ a j ě τ m by Lemma 2.3 (i), it follows that ∆ j ě δ τ , and hence C j ď ∆ ´ 2 ϵ { τ j ď ∆ ´ ϵ F j , (3.4) provided that δ ą 0 is chosen sufficiently small. Therefor e all applications of Theorem 1.1 below are legitimate with uniform constants depending only on s, t, u, ζ , η . In particular , Theorem 1.1 itself contributes no extra small losses in what follows; the only explicit losses come from the initial extraction of a large uniform subset and from comparing F and β via Remark 2.4 . For the intervals r ∆ a j ` 1 , ∆ a j s with j P I big 1 , thus t j P r 2 η , s s , we apply Theor em 1.1 with γ p 1 q j : “ t j ´ η ă t j , and for j P I small 1 we use the trivial bound | T p j | M j Á 1 . Therefor e, using Remark 2.4 , ź j P I 1 | T p j | M j Ç δ ź j P I big 1 ∆ p a j ´ a j ` 1 qp t j ´ η q Ç δ ź j P I 1 ∆ p a j ´ a j ` 1 qp t j ´ 2 η q ` Ç δ ∆ ´ F p A 1 q` 2 η A 1 ( 3.2 ) Ç δ ξ ∆ ´ β p A 1 q` 2 η A 1 . (3.5) For the intervals r ∆ a j ` 1 , ∆ a j s with j P I 2 , we apply Theorem 1.1 with γ p 2 q j : “ s ` t j 2 ´ η ă s ` t j 2 , 10 TUOMAS ORPONEN AND P ABLO SHMERKIN and Remark 2.4 , to get ź j P I 2 | T p j | M j Ç δ ź j P I 2 ∆ p a j ´ a j ` 1 q t j 2 ¨ ∆ s p a j ´ a j ` 1 q 2 ¨ ∆ ´ η p a j ` 1 ´ a j q Ç δ ∆ ´ F p A 2 q´ F p A 1 q 2 ¨ ∆ ´ s p A 2 ´ A 1 q 2 ¨ ∆ η p A 2 ´ A 1 q ( 3.2 ) Ç δ ξ { 2 ¨ ∆ ´ β p A 2 q´ β p A 1 q 2 ¨ ∆ ´ s p A 2 ´ A 1 q 2 ¨ ∆ η p A 2 ´ A 1 q . For the intervals r ∆ a j ` 1 , ∆ a j s with j P I 3 , we apply Theorem 1.1 with γ p 3 q : “ 1 ´ η ă 1 to get ź j P I 3 | T p j | M j Ç δ ź j P I 3 ∆ p a j ´ a j ` 1 qp 1 ´ η q “ ∆ A 2 ´ m ` η p m ´ A 2 q . (3.6) Combining the three cases, and using that δ ´ t 2 “ ∆ ´ 1 2 β p m q and ∆ η A 1 ě ∆ η m “ δ η , | T | M Ç δ δ ´ t 2 ` 2 η ` 3 ξ 2 ” ∆ ´ β p A 1 q 2 ¨ ∆ s p A 1 ´ A 2 q 2 ¨ ∆ p A 2 ´ m q` 1 2 p β p m q´ β p A 2 qq ı . (3.7) The inequality was written in this way to make explicit that all the powers of " ∆ " are non-positive. Let A : “ p 1 ´ t 2 q m . W e now divide the argument into cases, depending on the sizes of A 1 , A 2 relative to A . Cases A 1 ě A or A 2 ď A . The case A 1 ě A is nearly trivial: in this case we only need to observe, by the monotonicity of β , that ∆ ´ β p A 1 q 2 ě ∆ ´ β p A q 2 ( 3.3 ) ě δ ´ u ´ ϵ 2 . Plugging this into ( 3.7 ) leads to a better estimate than what we claimed. Let us then consider the case A 2 ď A . Now the main observation is that since β is 2 -Lipschitz, we have 1 2 β p A 2 q ě 1 2 β p A q ´ p A ´ A 2 q ( 3.3 ) ě p u ´ ϵ q m 2 ´ p A ´ A 2 q . Consequently , recalling that β p m q “ mt , we get ∆ p A 2 ´ m q` 1 2 p β p m q´ β p A 2 qq “ ∆ A 2 ´ A ´ 1 2 β p A 2 q ě ∆ ´ p u ´ ϵ q m 2 “ δ ´ u ´ ϵ 2 . Again, plugging this into ( 3.7 ) leads to a better estimate than what we claimed. FURSTENBERG TYPE ESTIMA TES 11 Case A 1 ď A ď A 2 . This is the main case. W e split the factor in ( 3.7 ) as ∆ ´ β p A 1 q 2 ¨ ∆ s p A 1 ´ A 2 q 2 ¨ ∆ p A 2 ´ m q` 1 2 p β p m q´ β p A 2 qq “ ∆ ´ β p A 1 q 2 ¨ ∆ s p A 1 ´ A q 2 ¨ ∆ s p A ´ A 2 q 2 ¨ ∆ A 2 ´ A ´ 1 2 β p A 2 q “ Π 1 ¨ Π 2 ¨ Π 3 ¨ Π 4 . It is desirable to combine the factors Π 3 and Π 4 : Π 3 ¨ Π 4 “ ∆ p A 2 ´ A qp 1 ´ s 2 q´ 1 2 β p A 2 q . (3.8) W e note that since β p m q “ mt , and β is 2 -Lipschitz, we have 1 2 p mt ´ β p A 2 qq ď m ´ A 2 ð ñ A 2 ´ A ď 1 2 β p A 2 q . Plugging this into ( 3.8 ), and using A 2 ě A , we find Π 3 ¨ Π 4 “ ∆ rp A 2 ´ A qp 1 ´ s 2 q´p 1 ´ s 2 q 1 2 β p A 2 qs´ s 4 β p A 2 q ě ∆ ´ β p A 2 q s 4 ě ∆ ´ β p A q s 4 ( 3.3 ) ě ∆ ´ ms p u ´ ϵ q 4 “ δ ´ s p u ´ ϵ q 4 . W e can obtain a similar estimate for the pr oduct Π 1 ¨ Π 2 by using the trivial estim- ate 1 2 β p A 1 q ě 1 4 β p A 1 q s , the 2 -Lipschitz property of β , and that A 1 ď A : Π 1 ¨ Π 2 “ ∆ ´ β p A 1 q 2 ¨ ∆ s p A 1 ´ A q 2 ě ∆ ´ β p A 1 q s 4 ¨ ∆ p β p A 1 q´ β p A qq s 4 “ ∆ ´ β p A q s 4 ( 3.3 ) ě δ ´ s p u ´ ϵ q 4 . All in all, Π 1 ¨ Π 2 ¨ Π 3 ¨ Π 4 ě δ ´ s p u ´ ϵ q 2 . Plugging this into ( 3.7 ), we obtain | T | M Ç δ δ ´ Λ , Λ : “ t 2 ` s p u ´ ϵ q 2 ´ 2 η ´ 3 ξ 2 . By the choices of η , ϵ, ξ , Λ “ t 2 ` su 2 ´ ´ sϵ 2 ` 2 η ` 3 ξ 2 ¯ ą t 2 ` su 2 ´ ω “ ζ . Therefor e, for δ small enough (depending on s, t, u, ζ ), the claimed bound ( 1.2 ) follows. □ 12 TUOMAS ORPONEN AND P ABLO SHMERKIN 4. P R O O F O F T H E O R E M S 1 . 4 A N D 1 . 5 W e first prove Theorem 1.4 . The proof of Theorem 1.5 will only differ in the very final computations, and we will indicate the necessary changes. Fix s P p 0 , 1 s , t P r s, 2 ´ s s , and η ą 0 . Let P Ă r 0 , 1 s 2 be a Katz-T ao p δ, t, δ ´ ϵ q -set. The claim is that if δ, ϵ ą 0 ar e suf ficiently small in terms of s, t, η , then | T | ě δ η M | P | p s ` t q{p 2 t q . (4.1) The "management of small constants" in these proofs is similar to that in the proof of Theorem 1.2 . There ar e three main parameters: the given η ą 0 , then the ϵ ą 0 we will choose, and finally an intermediate parameter ζ P p ϵ, η q , which is roughly the same as " ϵ F " in the proof of Theorem 1.2 . Since adding in all the small constants (again) makes the argument look complicated, we opt here for a notationally lighter strategy: we allow the Æ δ notation to hide constants of the form δ ´ o ζ Ñ 0 p 1 q . In this way , our final lower bound will have the form ě δ o ζ Ñ 0 p 1 q M | P | p s ` t q{p 2 t q , so we obtain ( 4.1 ) by choosing ζ ą 0 small enough in terms of η . The required smallness of ζ will finally determine the threshold for ϵ ! ζ , in fact via the relation ( 4.2 ). W e then start the proof in earnest. First of all, we may assume that P is a Katz- T ao p δ, t, 1 q -set, since the Katz-T ao p δ, t, δ ´ ϵ q -set P contains a Katz-T ao p δ, t, 1 q - subset P 1 with cardinality | P 1 | Ç δ | P | , see [ 6 , Lemma 2.2]. As in the proof of Theorem 1.2 , we may assume that δ “ ∆ m and P is t ∆ j u m j “ 1 - uniform for ∆ „ ϵ 1 . Let β : r 0 , m s Ñ r 0 , 2 m s be the branching function of P , defined in Section 2.2 , and apply Lemma 2.3 to β , with parameters d “ 2 and ζ . W e select ϵ ą 0 (depending on ζ , therefor e finally η ) so small that 2 ϵ { τ p ζ q ď ζ , (4.2) where τ p ζ q ą 0 is the parameter given by Lemma 2.3 . Let t a j u n j “ 0 , t t j u n ´ 1 j “ 0 be the objects given by Lemma 2.3 ; then n “ O ζ p 1 q by Lemma 2.3 (i), and a n “ m . The hypothesis that P is p δ, t, 1 q -Katz-T ao implies the uniform upper bound t j ď t, and in particular t j ď 2 ´ s because t ď 2 ´ s . W e apply Corollary 2.5 to the sequence ∆ a j , j “ n, n ´ 1 , . . . , 0 . W ith the nota- tion of the corollary , we bound | T | M Ç δ n ´ 1 ź j “ 0 | T p j | M j . (4.3) The implicit constant in the lower bound ( 4.3 ) is of the form Á p log p 1 { δ qq ´ O ζ p 1 q , and in particular it is larger than δ η { 2 if δ ą 0 is chosen suf ficiently small. Further in the notation of Corollary 2.5 , p S p j p P X p j q , T p j q is a p ∆ j , s, C p j , M j q - nice configuration with ∆ j : “ ∆ a j ` 1 ´ a j , and C p j À p log p 1 { δ qq O ϵ p 1 q δ ´ ϵ . In particular , FURSTENBERG TYPE ESTIMA TES 13 since a j ` 1 ´ a j ě τ m accor ding to Lemma 2.3 , and by ( 4.2 ), it holds C p j ď ∆ ´ 2 ϵ { τ j ď ∆ ´ ζ j , (4.4) provided that δ ą 0 was chosen suf ficiently small. W e split the indices t 0 , . . . , n ´ 1 u into 2 classes: I 1 “ t j P t 0 , . . . , n ´ 1 u : t j P r 0 , s su , I 2 “ t j P t 0 , . . . , n ´ 1 u : t j P p s, 2 ´ s su . Recall that the numbers t j are incr easing, and a n “ m . Let r 0 , a s “ Y j P I 1 r a j , a j ` 1 s , r a , m s “ Y j P I 2 r a j , a j ` 1 s , with the convention that a “ 0 if I 1 “ H and a “ m if I 2 “ H . For j P I 1 , we apply Theorem 1.1 to the configuration p S p j p P X p j q , T p j q at scale ∆ j , and with t j ď s . Note that S p j p P X p j q is p ∆ j , t j , O ϵ p 1 qq -Frostman thanks to the p t j , 0 q -superlinearity of p β , a j , a j ` 1 q , see [ 7 , Lemma 8.3] for the details. The conclusion is ź j P I 1 | T p j | M j Ç δ ź j P I 1 ∆ ´ t j j Ç δ ∆ ´ β p a q “ | P | ∆ a . (4.5) In this bound it was crucial that the "niceness" constant of the configuration p S p j p P X p j q , T p j q is bounded from above by ∆ ´ ζ j thanks to ( 4.4 ). This guaran- tees that the implicit constant in ( 4.5 ) is indeed of the form ě δ o ζ Ñ 0 p 1 q . Let us also mention that the middle estimate Ç δ ∆ ´ β p a q in ( 4.5 ) also uses Remark 2.4 in the same as the estimate ( 3.5 ) in the proof of Theor em 1.2 . For j P I 2 , we similarly apply Theorem 1.1 with t j P r s, 2 ´ s s and Remark 2.4 , to get ź j P I 2 | T p j | M j Ç δ ź j P I 2 ∆ ´p s ` t j q{ 2 j Ç δ ∆ ´ β p m q´ β p a q 2 ∆ ´ s p m ´ a q 2 “ | P | 1 { 2 ∆ a Ñ δ ¨ ˆ ∆ a δ ˙ s { 2 . (4.6) Here | P | ∆ a Ñ δ “ | P |{| P | ∆ a , and ˆ ∆ a δ ˙ s { 2 ě | P | s {p 2 t q ∆ a Ñ δ thanks to the Katz-T ao p δ, t, 1 q -set property of P . Combining the bounds from ( 4.5 )-( 4.6 ), and noting that 1 { 2 ` s {p 2 t q ď 1 by s ď t , we obtain | T | M Ç δ | P | ∆ a | P | 1 { 2 ` s {p 2 t q ∆ a Ñ δ ě | P | 1 { 2 ` s {p 2 t q ∆ a | P | 1 { 2 ` s {p 2 t q ∆ a Ñ δ “ | P | p s ` t q{p 2 t q . This completes the proof of Theor em 1.4 . W e next discuss the modifications needed to prove Theorem 1.5 . Let A, B Ă δ Z X r 0 , 1 s as in the statement of Theorem 1.5 . Then P : “ A ˆ B satisfies the hypotheses of Theorem 1.4 with t : “ α ` β , so we may repeat the pr oof to obtain the estimates ( 4.5 )-( 4.6 ). Consequently , | T | M Ç δ | P | ∆ a | P | 1 { 2 ∆ a Ñ δ ¨ ˆ ∆ a δ ˙ s { 2 “ | A | ∆ a | B | ∆ a | A | 1 { 2 ∆ a Ñ δ | B | 1 { 2 ∆ a Ñ δ ¨ ˆ ∆ a δ ˙ s { 2 . (4.7) 14 TUOMAS ORPONEN AND P ABLO SHMERKIN One minor differ ence, compared to previous argument, is that in the proof of Theorem 1.4 , the given set P was initially replaced by a t ∆ j u -uniform subset of nearly comparable cardinality . Here we rather replace both A and B individually by t ∆ j u -uniform subsets A 1 , B 1 , and then define P : “ A 1 ˆ B 1 . Evidently P is then also t ∆ j u -uniform. Defining P as a product set ensur es that | P | ∆ a “ | A 1 | ∆ a | B 1 | ∆ a and | P | ∆ a Ñ δ “ | A 1 | ∆ a Ñ δ | B 1 | ∆ a Ñ δ , which was needed in ( 4.7 ). T o proceed, fix θ P r 0 , 1 s satisfying max t θ s { α, p 1 ´ θ q s { β u ď 1 . (4.8) W e will first obtain a θ -dependent bound, and finally optimise θ to prove The- orem 1.5 . By the Katz-T ao p δ, α q -set and p δ , β q -set properties of A and B , ˆ ∆ a δ ˙ s { 2 “ ˆ ∆ a δ ˙ θs { 2 ¨ ˆ ∆ a δ ˙ p 1 ´ θ q s { 2 ě | A | θs {p 2 α q ∆ a Ñ δ | B | p 1 ´ θ q s {p 2 β q ∆ a Ñ δ . Plugging this lower bound into ( 4.7 ), and using 1 ě max t 1 2 ` θ s {p 2 α q , 1 2 ` p 1 ´ θ q s {p 2 β qu , we conclude | T | M Ç δ | A | 1 { 2 ` θs {p 2 α q ∆ a | B | 1 { 2 `p 1 ´ θ q s {p 2 β q ∆ a | A | 1 { 2 ` θs {p 2 α q ∆ a Ñ δ | B | 1 { 2 `p 1 ´ θ q s {p 2 β q ∆ a Ñ δ “ p| A || B |q 1 { 2 | B | s {p 2 β q ¨ p| A | s {p 2 α q | B | ´ s {p 2 β q q θ . It remains to optimise the value of θ under the constraints θ P r 0 , 1 s and ( 4.8 ). The assumption | A | β ě | B | α is equivalent to | A | s {p 2 α q | B | ´ s {p 2 β q ě 1 , so it is desirable to maximise θ . Remark 4.1. Actually the hypothesis | A | β ě | B | α is not formally needed her e – or even in Theorem 1.5 – but it shows why we want to choose θ as large as possible. In the opposite case | A | β ă | B | α we would instead minimise θ to obtain differ ent bounds. W e have not explicitly recor ded these bounds anywhere, since they can be obtained by swapping the roles of A, B in Theorem 1.5 . If α ď s , we choose θ : “ α { s P r 0 , 1 s to find | T | M Ç δ | A || B | p β ` s ´ α q{p 2 β q . (Note that the condition p 1 ´ θ q s { β “ p s ´ α q{ β ď 1 is guaranteed by s ď α ` β .) If α ě s , we instead choose θ : “ 1 to fi nd | T | M Ç δ | A | p α ` s q{p 2 α q | B | 1 { 2 . This concludes the proof of Theor em 1.5 . FURSTENBERG TYPE ESTIMA TES 15 5. S H A R P N E S S O F M A I N R E S U LT S In this section we verify the sharpness of our main r esults. W e start by recor d- ing the standard construction underlying the sharpness of the projection estim- ate corresponding to Theorem 1.1 . This construction goes back at least to [ 16 ]; see also [ 4 , §2.2, Case 2] and [ 1 , Section A.1]. In short, P 0 is a union of „ ρ ´ t many ρ -balls arranged on a product grid, while Θ 0 is obtained from arcs centr ed at rationals with small denominators. Lemma 5.1 (Standard sharp projection example) . For every dyadic scale ρ P p 0 , 1 s , every s P p 0 , 1 s , and every t P r s, 2 ´ s s , ther e exist: ‚ a set of the form P 0 “ A ˆ A Ă ρ Z 2 X r 0 , 1 s 2 , where A Ă ρ Z X r 0 , 1 s is a Frostman and Katz-T ao p ρ, t { 2 q -set with | A | „ ρ ´ t { 2 , ‚ a Frostman and Katz-T ao p ρ, s q -set Θ 0 Ă S 1 with | Θ 0 | ρ „ ρ ´ s , such that | π θ p P 0 q| ρ À ρ ´p s ` t q{ 2 , θ P Θ 0 . The Frostman and Katz-T ao constants in the claims above ar e absolute. Moreover , the elements of P 0 are Á ρ t { 2 -separated. 5.1. Sharpness of Theorem 1.2 and Corollary 1.3 . It is enough to establish the sharpness of Corollary 1.3 , since the corresponding sharpness for Theorem 1.2 then follows by the standard pr ojection/tube dictionary . Fix parameters s, t, u as in Cor ollary 1.3 , and let δ ą 0 be small. Choose dyadic scales ∆ „ δ u and ρ „ δ 1 ´ t { 2 ´ u { 2 , so that δ 1 ´ t { 2 „ ρ ∆ 1 { 2 and ρ 2 ∆ δ ´ 2 „ δ ´ t . Apply Lemma 5.1 with the parameter t “ 1 at scale ∆ . Thus there exist a set P 1 “ A 1 ˆ A 1 Ă ∆ 1 { 2 Z 2 X r 0 , 1 s 2 which is p ∆ , 1 , O p 1 qq -Fr ostman, satisfies | P 1 | ∆ „ ∆ ´ 1 , and, by the separation conclusion of the lemma, has the pr operty that every ∆ 1 { 2 -square meets P 1 in O p 1 q many ∆ -squar es. Moreover , ther e exists a Frostman p ∆ , s q -set of slopes Θ 1 such that | π θ p P 1 q| ∆ À ∆ ´p 1 ` s q{ 2 , θ P Θ 1 . Now define P 0 : “ ρ P 1 (scaling by ρ ). Then | P 0 | δ „ δ ´ t , and because δ 1 ´ t { 2 „ ρ ∆ 1 { 2 , every δ 1 ´ t { 2 -square contains at most O p ∆ q -proportion of P 0 , i.e. | P 0 X Q | δ À δ u | P 0 | δ , Q P D δ 1 ´ t { 2 . Next modify Θ 1 to a Frostman p δ , s, O p 1 qq -set Θ 0 by replacing each ∆ -interval in Θ 1 by „ p ∆ { δ q s many maximally spaced δ -intervals. If θ 1 P Θ 0 lies above θ P Θ 1 , then | θ 1 ´ θ | À ∆ , so | π θ 1 p P 0 q| ρ ∆ À | π θ p P 0 q| ρ ∆ À ∆ ´p 1 ` s q{ 2 . Passing from scale ρ ∆ to scale δ gives, for every θ 1 P Θ 0 , | π θ 1 p P 0 q| δ À ρ ∆ δ ¨ ∆ ´p 1 ` s q{ 2 „ δ ´p t { 2 ` su { 2 q . 16 TUOMAS ORPONEN AND P ABLO SHMERKIN Since the same upper bound holds a fortiori for every subset P 1 Ă P 0 , the expo- nent in Corollary 1.3 is sharp. 5.2. Sharpness of Theorem 1.4 . Fix s P p 0 , 1 s , t P r s, 2 ´ s s , and δ P 2 ´ N X p 0 , 1 s . Choose ρ P 2 ´ N X r δ, 1 s , and apply Lemma 5.1 at scale ρ . Then there exist a set P 0 and a Frostman p ρ, s q -set Θ 0 as in the lemma, with | P 0 | „ ρ ´ t and | π θ p P 0 q| ρ À ρ ´p s ` t q{ 2 „ | P 0 | p s ` t q{p 2 t q , θ P Θ 0 . Rescaling by the factor δ { ρ , we obtain a Katz-T ao p δ , t, O p 1 qq -set P : “ p δ { ρ q P 0 with | P | “ | P 0 | „ ρ ´ t and | π θ p P q| δ À ρ ´p s ` t q{ 2 „ | P | p s ` t q{p 2 t q , θ P Θ 0 . Passing to the ρ -neighbourhood Θ of Θ 0 preserves the same upper bound up to constants and yields a Fr ostman p δ, s q -set of directions. Since ρ can be chosen dyadically anywhere in r δ , 1 s , the car dinality | P | „ ρ ´ t ranges through the whole admissible interval r 1 , δ ´ t s up to constants. This shows that the lower bound in Theorem 1.4 is sharp thr oughout the admissible range of cardinalities. 5.3. Sharpness of Theorem 1.5 . W e finally discuss the sharpness of Theor em 1.5 for product sets, but only in the special case α “ β , and 2 α P r s, 2 ´ s s , as men- tioned in Remark 1.6 . In that case, recall that Theor em 1.5 yields the bounds | T | ě δ η M # | A || B | s {p 2 α q , α ď s, | A | 1 { 2 ` s {p 2 α q | B | 1 { 2 , α ě s, (5.1) for | A | ě | B | (this is also true but sub-optimal for | A | ă | B | ). Let us first consider the sharpness of ( 5.1 ) when α ď s . Fix the "target car din- alities" | A | ě | B | . The aim is to construct Katz-T ao p δ, α q -sets A, B Ă r 0 , 1 s with these car dinalities, and a family of tubes T , such that p A ˆ B , T q is a p δ , s, C, δ ´ s q - nice configuration, and the first bound in ( 5.1 ) is attained. Recall that the symmetric sharp example from Lemma 5.1 (with t : “ 2 α ) has the form B 0 ˆ B 0 . Let ℓ : “ δ | B | 1 { α , choose B 0 Ă δ Z X r 0 , ℓ s to be a Katz-T ao p δ, α q -set with | B 0 | “ | B | , choose a Katz-T ao p ℓ, α q -set D Ă ℓ Z X r 0 , 1 ´ ℓ s with | D | „ N : “ | A |{| B | (this is possible since | A | À δ ´ α , hence N À p δ | B | 1 { α q ´ α “ ℓ ´ α ), and define A 0 : “ D ` B 0 . Then | A 0 | “ N | B 0 | „ | A | , and A 0 is again a Katz-T ao p δ, α q -set: if I is an interval of length r ď ℓ , it meets at most O p 1 q translates d ` B 0 , while for r ě ℓ , it meets at most O pp r { ℓ q α q such translates, each contributing at most | B 0 | „ p ℓ { δ q α points. Hence A 0 ˆ B 0 is a union of N disjoint translates of B 0 ˆ B 0 , and for the r esulting configuration one has | T | À M N ¨ | B | p s ` 2 α q{p 2 α q „ M | A || B | s {p 2 α q . This matches the first bound in ( 5.1 ). FURSTENBERG TYPE ESTIMA TES 17 W e then discuss the sharpness of the second bound in ( 5.1 ). This is based on the following discrete statement: Lemma 5.2. For A , B , C P N z t 0 u satisfying A ě t B , C u and A ď BC , (5.2) there exist sets A, B , C Ă r 0 , 1 s of cardinalities comparable to A , B , C , such that | A ` cB | À ? ABC , c P C. Moreover , the sets A, B , C are maximally spaced: | A | A ´ 1 „ A , | B | B ´ 1 „ B and | C | C ´ 1 „ C . (5.3) Remark 5.3. This lemma is closely related to the "sharpness of optimal the AB C sum-product theor em" in [ 5 , Theorem 1.2]. Since [ 5 ] does not appear to state the precise discr ete claim useful for us, we give an elementary argument below . Proof. Let A Ă r 0 , 1 s be a maximal arithmetic progr ession (AP) containing 0 with gap A ´ 1 . Next, let B Ă A be a maximal sub-AP with 0 P B and gap on the interval r 1 2 B ´ 1 , B ´ 1 s . This exists, because A { B ě 1 , so there exists a natural number m P r 1 2 A { B , A { B s . W ith this notation the gap of B could be m A ´ 1 . Let a P A and b P B be the largest elements with a ď a C { AB and b ď a C { AB , respectively . Note that a ě b ą 0 since B Ă A , and B ´ 1 ď a C { AB thanks to ( 5.2 ). Using that B is an AP , b ą 0 implies that a fortiori a, b „ a C { AB , and | B X r 0 , b s| „ b B „ a BC { A . W rite I : “ r 0 , a s Ă r 0 , a C { AB s ( 5.2 ) Ă r 0 , 1 s , and A I : “ A X I , and B I : “ B X I “ B X r 0 , b s . Let C consist of rationals of the form a { b P r 0 , 1 s , where a P A I ´ A I and b P p B I ´ B I q z t 0 u . W e claim that | C | C ´ 1 „ C . (5.4) T o see this, note first that A 1 : “ t A a : a P A X I u “ t 0 , . . . , A a u consists of all natural numbers between 0 and n : “ A a . Second, B 1 : “ t A b : b P B X I u Ă t 0 , . . . , n u is an arithmetic progr ession with | B 1 | “ | B I | „ a BC { A , and D : “ diam p B 1 q „ A b „ a A C { B . Now applying Corollary A.3 to A 1 and B 1 shows that ˇ ˇ ˇ r 0 , 1 s X A 1 ´ A 1 p B 1 ´ B 1 q z t 0 u ˇ ˇ ˇ p| B 1 | D q ´ 1 „ | B 1 | D „ C . This implies ( 5.4 ), since A 1 , B 1 are scaled copies of A I , B I with common factor A . 18 TUOMAS ORPONEN AND P ABLO SHMERKIN Finally , we claim that | A ` cB | À ℓ p I q| A || B | „ ? ABC , c P C, where ℓ p I q is the length of I . T o see this, fix c “ p a 1 ´ a 2 q{p b 1 ´ b 2 q and r P A ` cB , where a j P A X I and b j P B X I . In particular max t a j , b j u ď ℓ p I q . W e will show that |tp a, b q P p A ` A ´ A q ˆ p B ´ B ` B q : a ` cb “ r u| Á ℓ p I q ´ 1 (5.5) for all r P A ` cB , which yields the desired estimate | A ` cB | À ℓ p I q|p A ` A ´ A q ˆ p B ´ B ` B q| À ℓ p I q| A || B | . T o prove ( 5.5 ), fix r P A ` cB , and let p a 0 , b 0 q P A ˆ B with a 0 ` cb 0 “ r . Recalling that c “ p a 1 ´ a 2 q{p b 1 ´ b 2 q , also p a 0 ` ma 1 ´ ma 2 q ` c p b 0 ´ mb 1 ` mb 2 q “ r , m P Z . For m ď ℓ p I q ´ 1 , it holds max t ma j , mb j u ď 1 . Thus, for m ď ℓ p I q ´ 1 , the points ma j are elements of the AP A Ă r 0 , 1 s (the maximal AP on r 0 , 1 s with gap A ´ 1 ) and the points mb j are similarly elements of B Ă r 0 , 1 s . Therefor e a 0 ` ma 1 ´ ma 2 P A ` A ´ A and b 0 ´ mb 1 ` mb 2 P B ´ B ` B for m ď ℓ p I q ´ 1 , giving ( 5.5 ). □ Now we get back to the sharpness of the second bound in ( 5.1 ). Let | A | , | B | P r 1 , δ ´ α s with | B | ď | A | and | B | α ě | A | s ´ α be the "target cardinalities" for which we want to establish the sharpness of ( 5.1 ). The goal is to construct Katz-T ao p δ, α q -sets A, B Ă r 0 , 1 s with roughly these cardinalities, and a Frostman p δ, s q -set Θ Ă S 1 such that | π θ p A ˆ B q| À | A | 1 { 2 ` s {p 2 α q | B | 1 { 2 for all θ P Θ . T o be precise, we will construct a Fr ostman p δ, s q -set C Ă r 0 , 1 s such that | A ` cB | δ À | A | 1 { 2 ` s {p 2 α q | B | 1 { 2 , c P C. (5.6) Let A 0 , B 0 , C 0 Ă r 0 , 1 s be the sets fr om Lemma 5.2 with the choices A : “ | A | , B : “ | B | , and C : “ | A | s { α . Recalling that s ď α , one may check that the requirements ( 5.2 ) are met; in par- ticular A ď BC is equivalent to | B | α ě | A | s ´ α . Then ( 5.3 ) promises that e.g. C 0 contains an | A | ´ s { α -separated subset of cardinality „ | C 0 | „ | A | s { α . W e reduce C 0 to this subset, so we assume in the sequel that C 0 is | A | ´ s { α -separated. W rite ∆ : “ δ | A | 1 { α , and let A : “ ∆ A 0 and B : “ ∆ B 0 . Using | B | ď | A | and ( 5.3 ), it is easy to check that A, B are Katz-T ao p δ, α q -sets. Moreover , | A ` cB | δ ď | A 0 ` cB 0 | | A | ´ 1 { α À a | A 0 || B 0 || C 0 | „ | A | 1 { 2 ` s {p 2 α q | B | 1 { 2 , c P C 0 . (5.7) Evidently ( 5.7 ) persists for all c P C , where C is the | A | ´ 1 { α -neighbourhood of C 0 . Therefor e we have established ( 5.6 ) for A, B , and C . It remains is to check that C is a Frostman p δ , s q -set. FURSTENBERG TYPE ESTIMA TES 19 First recall that C 0 is | A | ´ s { α -separated and | C 0 | „ | A | s { α . Since s ď 1 , it follows that | C | δ „ | A | ´ 1 { α δ ¨ | C 0 | „ | A | p s ´ 1 q{ α δ ´ 1 . Now , for I P D r pr 0 , 1 sq with δ ď r ď | A | ´ 1 { α , we have r 1 ´ s ď | A | p s ´ 1 q{ α , hence | C X I | δ ď p r { δ q ď r s | A | p s ´ 1 q{ α δ ´ 1 “ r s | C | δ . For | A | ´ 1 { α ď r ď | A | ´ s { α we use the | A | ´ s { α -separation of C 0 to estimate | C X I | δ À | A | ´ 1 { α δ ď r s | A | p s ´ 1 q{ α δ ´ 1 „ r s | C | δ . Finally , for | A | ´ s { α ď r ď 1 , we use both the | A | ´ s { α -separation of C 0 , and the Frostman p| A | ´ s { α , s q -set pr operty of C 0 to estimate | C X I | δ À | A | ´ 1 { α δ ¨ | C 0 X I | ď | A | ´ 1 { α δ ¨ r s | C 0 | „ r s | C | δ . W e have now verified that C is a Frostman p δ, s q -set, and established the sharp- ness of the second bound in ( 5.1 ). A P P E N D I X A. R AT I O S O F A R I T H M E T I C P R O G R E S S I O N S This appendix contains some (with very high likelihood) standard facts which were needed to prove Lemma 5.2 . W e start with the following variant of Dirich- let’s approximation theor em: Lemma A.1. Assume that m, n P N z t 0 u are such that m ď n , and m divides n . Write A “ t 0 , 1 , . . . , n u and B “ t 0 , m, 2 m, . . . , n u . Then, for all x P r 0 , 1 s , there exist a P A ´ A and b P p B ´ B q z t 0 u such that | x ´ a b | ď m | b | n . Proof. Fix x P r 0 , 1 s . For every k P t 0 , . . . , n { m u we may write k mx “ n k ` x k , where n k P t 0 , . . . , k m u Ă A and x k P r 0 , 1 q . There ar e n { m ` 1 many numbers x k , so there is a pair x i , x j with i ‰ j , and | x i ´ x j | ď m { n . Consequently , |p i ´ j q mx ´ p n i ´ n j q| “ | x i ´ x j | ď m { n, which can be rearranged to ˇ ˇ ˇ ˇ x ´ n i ´ n j im ´ j m ˇ ˇ ˇ ˇ ď m | im ´ j m | n . This gives the claim with a “ n i ´ n j and b “ im ´ j m . □ The following proposition is an asymmetric variant of [ 1 , Claim A.5] Proposition A.2. Let A, B be as in Lemma A.1 . Then ˇ ˇ ˇ ˇ r 0 , 1 s X A ´ A p B ´ B q z t 0 u ˇ ˇ ˇ ˇ p| A || B |q ´ 1 Á | A || B | “ n 2 m . 20 TUOMAS ORPONEN AND P ABLO SHMERKIN Proof. W e rewrite the conclusion of Lemma A.1 as follows: r 0 , 1 s Ă ď b Pp B ´ B q z t 0 u ď a P A ´ A B ˆ a b , m | b | n ˙ . W e may sharpen this a little. If b P p B ´ B q z t 0 u is fixed, all points a P A ´ A such that | x ´ a { b | ď p bn q ´ 1 for some x P r 0 , 1 s satisfy dist p a, b r 0 , 1 sq ď n ´ 1 . Thus a lies on a certain interval I b of length at most | b | ` n ´ 1 . T aking this into account, r 0 , 1 s Ă ď b Pp B ´ B q zt 0 u ď a Pp A ´ A qX I b B ˆ a b , m | b | n ˙ . Let us consider: how much of r 0 , 1 s can be covered by that part of the previous union where | b | ď cn ? Here c ą 0 is a suitable constant to be determined in a moment. T aking into account that card pp A ´ A q X I b q ď | b | ` 1 ď 2 | b | and card pt b P B ´ B : | b | ď cn uq À cn { m , the measur e of this "bad" part I bad Ă r 0 , 1 s is at most | I bad | ď ÿ b Pp B ´ B q z t 0 u | b |ď cn card pp A ´ A q X I b q ¨ 2 m | b | n À ÿ b Pp B ´ B q z t 0 u | b |ď cn 2 | b | ¨ 2 m | b | n À c. Thus, if c ą 0 is small enough, we have |r 0 , 1 s z I bad | ě 1 2 . Moreover , r 0 , 1 s z I bad Ă ď b Pp B ´ B q z t 0 u | b |ě cn ď a Pp A ´ A q B ˆ a b , m | b | n ˙ Ă ď b Pp B ´ B q z t 0 u | b |ě cn ď a Pp A ´ A q B ´ a b , m cn 2 ¯ . From this inclusion we see that 1 2 ď |r 0 , 1 s z I bad | À |t a b : p a, b q P p A ´ A q ˆ p B ´ B q z t 0 uu| m { n 2 ¨ m n 2 , which gives the claim. □ W e finally recor d a more applicable version without divisibility hypotheses: Corollary A.3. Let n P N , A : “ t 0 , . . . , n u , and let B Ă t 0 , . . . , n u be an arithmetic progr ession with D : “ diam p B q ą 0 . Then, ˇ ˇ ˇ ˇ r 0 , 1 s X A ´ A p B ´ B q z t 0 u ˇ ˇ ˇ ˇ p| B | D q ´ 1 Á | B | D . Proof. W rite B “ t b 0 , b 0 ` m 0 , b 0 ` 2 m 0 , . . . , b 1 u , where b 1 ´ b 0 “ D , and m 0 ě 1 is the gap of B . It suffices to show that ˇ ˇ ˇ ˇ r 0 , 1 s X p A X r b 0 , b 1 sq ´ p A X r b 0 , b 1 sq p B ´ B q z t 0 u ˇ ˇ ˇ ˇ p| B | D q ´ 1 Á | B | D FURSTENBERG TYPE ESTIMA TES 21 W rite n 0 : “ b 1 ´ b 0 P N z t 0 u , A 0 : “ p A X r b 0 , b 1 sq ´ b 0 “ t 0 , . . . , n 0 u , and B 0 : “ B ´ b 0 Ă t 0 , . . . , n 0 u . W ith this notation, p A X r b 0 , b 1 sq ´ p A X r b 0 , b 1 sq p B ´ B q z t 0 u “ A 0 ´ A 0 p B 0 ´ B 0 q z t 0 u . Now the gap " m 0 " of B (hence B 0 ) divides n 0 , so we may apply Proposition A.2 . Noting also that | A 0 | “ D and | B 0 | “ | B | , the conclusion is ˇ ˇ ˇ ˇ r 0 , 1 s X A ´ A p B ´ B q z t 0 u ˇ ˇ ˇ ˇ p| B | D q ´ 1 ě ˇ ˇ ˇ r 0 , 1 s X A 0 ´ A 0 p B 0 ´ B 0 q z t 0 u ˇ ˇ ˇ p| B | D q ´ 1 Á | B | D , as desired. □ R E F E R E N C E S [1] Damian D ˛ abrowski, T uomas Orponen, and Hong W ang. How much can heavy lines cover? J. Lond. Math. Soc. (2) , 109(5):Paper No. e12910, 33, 2024. [2] Ciprian Demeter and W illiam O’Regan. Incidence estimates for quasi-product sets and ap- plications. arXiv preprint , nov 2025. . [3] Ciprian Demeter and Hong W ang. Szemerédi-Trotter bounds for tubes and applications. Ars Inven. Anal. , pages Paper No. 1, 46, 2025. [4] Y uqiu Fu, Shengwen Gan, and Kevin Ren. An incidence estimate and a Furstenberg type estimate for tubes in R 2 . J. Fourier Anal. Appl. , 28(4):Paper No. 59, 28, 2022. [5] Longhui Li and Bochen Liu. Dimension of Diophantine approximation and some applica- tions in harmonic analysis. arXiv e-prints , page arXiv:2409.12826, September 2024. [6] T uomas Orponen. AB C sum-product theorems for Katz-T ao sets. arXiv pr eprint , November 2025. . [7] T uomas Orponen and Pablo Shmerkin. On the Hausdorff dimension of Furstenber g sets and orthogonal projections in the plane. Duke Math. J. , 172(18):3559–3632, 2023. [8] T uomas Orponen and Pablo Shmerkin. Projections, furstenberg sets, and the AB C sum-product problem. J. Amer . Math. Soc. , accepted for publication, 2026. Preprint: arXiv:2301.10199 . [9] T uomas Orponen and Guangzeng Y i. Large cliques in extremal incidence configurations. Rev . Mat. Iberoam. , 41(4):1431–1460, 2025. [10] Kevin Ren and Hong W ang. Furstenberg sets estimate in the plane. arXiv pr eprint , August 2023. . [11] Pablo Shmerkin. A non-linear version of Bourgain’s projection theorem. J. Eur . Math. Soc. (JEMS) , 25(10):4155–4204, 2023. [12] Pablo Shmerkin and Hong W ang. Dimensions of Furstenberg sets and an extension of Bour- gain’s projection theor em. Anal. PDE , 18(1):265–278, 2025. [13] Pablo Shmerkin and Hong W ang. On the distance sets spanned by sets of dimension d { 2 in R d . Geom. Funct. Anal. , 35(1):283–358, 2025. [14] Hong W ang and Shukun W u. Restriction estimates using decoupling theorems and two- ends Furstenberg inequalities. arXiv pr eprint , November 2024. . [15] Hong W ang and Shukun W u. T wo-ends Furstenber g estimates in the plane. arXiv preprint , September 2025. . [16] Thomas W olff. Recent work connected with the Kakeya pr oblem. In Prospects in mathematics (Princeton, NJ, 1996) , pages 129–162. Amer . Math. Soc., Providence, RI, 1999. 22 TUOMAS ORPONEN AND P ABLO SHMERKIN D E PA RT M E N T O F M AT H E M A T I C S A N D S TAT I S T I C S , U N I V E R S I T Y O F J Y V Ä S K Y L Ä , P . O . B O X 3 5 ( M A D ) , F I - 4 0 0 1 4 U N I V E R S I T Y O F J Y V Ä S K Y L Ä , F I N L A N D Email address : tuomas.t.orponen@jyu.fi D E PA RT M E N T O F M AT H E M AT I C S , T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A , 1 9 8 4 M AT H E M - AT I C S R O A D , V A N C O U V E R , B C , C A N A D A Email address : pshmerkin@math.ubc.ca