Fourier dimension of Mandelbrot Cascades on planar curves

F OURIER DIMENSION OF MANDELBR OT CASCADES ON PLANAR CUR VES DONGGEUN R YOU AND VILLE SUOMALA Abstract. W e consider multifractal Mandelbrot cascades supp orted on pla- nar C 2 curves with non v anishing curv ature and show that their F ourier dimen- sion is as large as p ossible, i.e., equal to the infimum of the low er p oin t wise dimension of the measure. 1. Introduction F or a finite measure η on an Euclidean space R d , we define the F ourier transform of η at ξ ∈ R d as b η ( ξ ) = Z R n e − 2 π ix · ξ dη ( x ) . While the study of Ra jchman measures, i.e. those whose F ourier transform v an- ishes at infinit y , dates bac k more than a century (see [14]), the problem of quan ti- fying the deca y rate of the F ourier transform for random measures w as initiated in the early 1970s b y Mandelbrot [15, 16]. This line of researc h was further brought up by Kahane [9], who revisited sev eral of the Mandelbrot problems a few decades later. Although Kaufman’s work in the early 1980s [11] established the existence of random measures with arbitrary prescrib ed F ourier dimension, it to ok almost 50 y ears before the first nontrivial results app eared for the models that Mandelbrot was in terested in. Before presenting these results, we recall that the F ourier dimension of a measure η , denoted b y dim F η , is defined as dim F η = sup n 0 < s < d : | b η ( ξ ) | = O ( | ξ | − s/ 2 ) o . In [5], F alconer and Jin pro vided the first quan titative low er bounds for the F ourier dimension of Gaussian multiplicativ e chaos (GMC) measures. Their re- sult concerns the GMC defined on planar domains, and the metho d relies on the theory of orthogonal pro jections. The corresp onding problem on the real line was addressed by Garban and V argas [6], who obtained a lo wer b ound for the F ourier dimension of the GMC on [0 , 1]. F or the related dyadic mo del of Mandelbrot cas- cades, the exact v alue of the F ourier dimension w as determined indep enden tly b y Chen, Han, Qiu, and W ang [4], and by Chen, Li, and Suomala [3]. Subsequently , the F ourier dimension of more general random measures, including GMC defined on the torus T d ⊂ R d , was established in [12, 13]. All the results discussed abov e concern random measures defined on domains in R d . In particular, for cascade measures ν on [0 , 1] d , the results of [3, 4] show that the F ourier dimension is, almost surely , given by dim F ν = min { 2 , dim 2 ν } , (1) Date : March 27, 2026. VS was supp orted by the Researc h Council of Finland via the pro ject “ F ractals and r andom- ness ”, grant no. 368817. W e thank T uomo Kuusi for p oin ting our attention to the concentration bounds in [1] and T uomas Sahlsten for useful discussions. 1 2 DONGGEUN R YOU AND VILLE SUOMALA where dim 2 η denotes the correlation dimension of a measure η , and whose v alue for the cascade measures has a well-kno wn explicit form ula. Recalling the general inequalit y dim F η ≤ dim 2 η , v alid for all finite Borel measures η , w e th us note that cascade measures on [0 , 1] d are quasi-Salem, in the sense that their F ourier dimension is as large as p ermitted b y this inequalit y . In this pap er, w e extend the study of Mandelbrot m ultiplicativ e cascades to cascade measures supported on planar curv es with non v anishing curv ature. This problem w as proposed in [4, p. 7] in the language of oscillatory in tegrals, and w as also discussed in [3], where a partial and non-optimal result w as obtained. Our main result sho ws that, for suc h cascade measures, the F ourier dimension is almost surely equal to the minimum of the multifractal sp ectrum. More precisely , let ( W i ) 1 ≤ i ≤ b b e non-negativ e random v ariables with unit expe ctation and superp olynomial tails. Let Γ ⊂ R 2 b e a compact C 2 curv e with nonv anishing curv ature, and let µ b e the Mandelbrot cascade constructed from ( W i ) b i =1 via a b -adic decomp osition of Γ. Denote α min = inf { dim( µ, x ) : x ∈ spt µ } , where dim( η , x ) is the p oint wise Hausdorff dimension of a measure η at x . Deferring the detailed definitions and our technical assumptions to Section 2.1, we now state the main theorem. Theorem 1.1. dim F µ = α min almost sur ely on non-extinction. W e note that α min has a w ell-kno wn analytic expression in terms of the random v ariables ( W i ). W e also extend the result by establishing b ounds on the deca y rate of the spherical L p a verages, σ p ( η )( r ) =  Z S 1 | b η ( rθ ) | p dσ ( θ )  1 /p , (2) for all 1 ≤ p ≤ ∞ , where σ denotes the surface measure on the unit circle. W e refer to Theorem 5.1 for the precise statement and note that Theorem 1.1 corresp onds to the case p = ∞ , interpreting the definition (2) as the L ∞ norm if p = ∞ . In addition, as a b yproduct of the proof of Theorem 1.1, w e obtain a proof of (1), assuming that ( W i ) i satisfies the same hypotheses as in Theorem 1.1. Although this result is contained in [13], our approac h yields a relatively simple proof under the fairly general moment conditions on the random weigh ts. In particular, it applies to lognormal cascades, whic h constitute the most relev an t examples from the viewp oin t of applications. The pap er is organized as follows. In Section 2.1, we introduce the mo del and recall the necessary m ultifractal to ols. In Section 2.2, we recall the form ula for the correlation dimension of the cascades and pro vide a technical lemma describing the b eha viour of certain momen t sums of the cascade across differen t scales. Section 2.3 con tains our main probabilistic ingredien t: a concentration inequalit y adapted from [1], which enables us to impro ve the methods developed in [3, 19]. This strategy is implemen ted in Section 3, where we establish the lo wer bound dim F µ ≥ α min . The matc hing upp er b ound dim F µ ≤ α min follo ws as a corollary of a universal estimate that holds for any finite Borel measure supp orted on a C 2 curv e with nonv anishing curv ature. The details are provided in Section 4. Finally , our result on the deca y of σ p ( µ )( r ) is presented in Section 5, and our new pro of of (1) is found in the App endix. 2. Preliminaries 2.1. Notation for cascade measures. In relation to the F ourier dimension, we define the correlation dimension, dim 2 η , of a measure η . This may b e done b y partitioning the space into cubical ob jects of a certain size, computing the L 2 -sum of their masses, and passing to the limit after a suitable normalisation. F or a FOURIER DIMENSION OF MANDELBROT CASCADES ON PLANAR CUR VES 3 measure η on R d , we let dim 2 η = lim inf n →∞ log P Q ∈Q n η ( Q ) 2 − n , where Q n is the collection of b -adic subsquares of side-length b − n and the log is to base b . Here and in what follows, b ≥ 2 is a fixed in teger. It is well kno wn that dim F η ≤ dim 2 η ≤ dim H η holds for all compactly supp orted finite measures, where dim H η is the Hausdorff dimension of η defined as the suprem um of the v alues s for whic h dim( η , x ) ≥ s holds for η -almost every x , and dim( η , x ) = lim inf r → 0 log( η ( B ( x, r )) log r is the point wise Hausdorff dimension of η at x . W e first define Mandelbrot mul- tiplicativ e cascades on [0 , 1] d with resp ect to the base b . T o that end, let Λ = { 0 , . . . , b − 1 } d , and given i = ( i 1 , . . . , i n ) ∈ Λ n , let x i ∈ [0 , 1] d suc h that ( x i ) j = P n k =1 ( i k ) j b − k for all 1 ≤ j ≤ d , and let Q i = x i + [0 , b − n ) d . Then Q n = { Q i : i ∈ Λ n } is the family of half-op en b -adic sub cub es of [0 , 1) d of level n . If i = ( i 1 , . . . , i n ) ∈ Λ n , and 1 ≤ j ≤ n , we use i | j = ( i 1 , . . . , i j ) to denote the sub word con taining the first j elements. The cascade is driven by a random v ector W = ( W i ) i ∈ Λ , where the W i ≥ 0 are random v ariables with E X i ∈ Λ W i ! = b d . (3) W e attach an indep enden t copy W i of W to each i ∈ Λ n , n ∈ N . F or a fixed i = ( i j ) n j =1 ∈ Λ n , let ν n ( x ) = n Y j =1 ( W i | j − 1 ) i j , for eac h x ∈ Q i (where W ∅ = W ), and let ν b e the weak*-limit of the measures dν n ( x ) = ν n ( x ). This random cascade measure ν is non-zero with p ositive proba- bilit y if and only if the condition X i ∈ Λ E ( W i log W i ) < db d (4) is satisfied [10]. This subcriticality condition (4) is our standing assumption through- out the paper. In our main result, we consider Mandelbrot m ultiplicative cascades defined on curv es with nonzero curv ature. In this curvilinear setting, the cascade measure is the push forw ard ν ◦ γ − 1 where γ : [0 , 1] → R 2 is a C 2 -curv e with det( γ ′ ( t ) , γ ′′ ( t ))  = 0, and ν is the cascade measure on the unit interv al. W e use the following notation: let Q n , n ≥ 0 b e the b -adic filtration on the unit interv al and let D n = γ ( Q n ). Let µ n = ν n ◦ γ − 1 , µ = ν ◦ γ − 1 , where ( ν n ) n is the sequence of the cascade measures asso ciated with an initial random v ariable W . Without loss of generality , w e assume | γ ′ | = 1 and denote Z γ ( J ) f ( x ) dx = Z J f ( γ ( t )) dt , so that the integration is with resp ect to the arc length. Let us also denote Γ = γ ([0 , 1]). Throughout the pap er, we impose the follo wing additional assumptions on W i , i ∈ Λ: E ( W i ) = 1 , (5) E ( W p i ) < ∞ for all 0 < p < ∞ . (6) 4 DONGGEUN R YOU AND VILLE SUOMALA Denote for q ≥ 0, τ ( q ) = dq − log X i ∈ Λ E ( W q i ) ! . If there exists a v alue q such that q τ ′ ( q ) = τ ( q ), then this v alue is unique. W e denote it by q max . Note that q τ ′ ( q ) ≥ τ ( q ) if and only if q ≤ q max . If q τ ′ ( q ) ≥ τ ( q ) for all q , w e let q max = ∞ . F or p > 0, w e define e τ ( p ) as follows: e τ ( p ) = ( τ ( p ) if τ ′ ( p ) ≥ τ ( p ) /p pτ ( q max ) q max otherwise. (7) Also, we let α min = τ ′ ( q max ) = τ ( q max ) q max , or let α min = lim q →∞ τ ′ ( q ) if q max = ∞ . In other w ords, α min = lim p →∞ e τ ( p ) p . Note that α min = inf { dim( ν, x ) : x ∈ spt ν } = inf { dim( µ, x ) : x ∈ spt µ } almost surely on non-extinction of the measure µ . W e refer e.g. to [18, 2, 7] for these and other basic facts ab out the m ultifractal analysis of the cascade measures. W e will use the standard O ( · ) notation and also the notation f ≲ g as a synon ym to f = O ( g ). If f ≲ g ≲ f , w e denote f ∼ g . If a constant C may dep end on a parameter such as p , w e write f ≲ p g meaning that f ≤ C ( p ) g where the constant C ( p ) may dep end on p . The dep endence on implicit constants will b e clarified as needed. F or ξ ∈ R d , we use the familiar notation | ξ | ∞ = max {| ξ 1 | , · · · , | ξ d |} . 2.2. Auxiliary results for the Mandelbrot cascades. Throughout this section, w e pro v e auxiliary results for the cascade measures. W e note that the results in this section do not depend on the geometry of the supp ort of the cascade, whether it is a cub e in R d or a curv e, and they could b e stated purely in symbolic terms via Λ N . T o cov er also the case d > 1, we state the results for cascades on [0 , 1] d , but w e stress that they also hold for the curvilinear cascades. F or 1 ≤ p, q < ∞ and 1 ≤ j ≤ n , we let S ( p, q , j, n ) =    X I ∈Q j   X J ∈Q n ,J ⊂ I ν n ( J ) q   p/q    1 /p . If p = ∞ , we take the ℓ ∞ norm ov er I ∈ Q j , i.e. S ( ∞ , q , j, n ) = sup I ∈Q j   X J ∈Q n ,J ⊂ I ν n ( J ) q   1 /q . Similarly , S ( p, ∞ , j, n ) =   X I ∈Q j  sup J ∈Q n ,J ⊂ I ν n ( J )  p   1 /p . If p = q = ∞ , we define S ( ∞ , ∞ , j, n ) := S ( ∞ , 1 , n, n ). T o estimate E ( S ( p, q, j, n )), w e use the auxiliary random v ariables W q = b d ( W q j ) j ∈ Λ P j ∈ Λ E ( W q j ) , FOURIER DIMENSION OF MANDELBROT CASCADES ON PLANAR CUR VES 5 and the auxiliary measure defined b y setting ν j,j ( x ) = 1, and ν j,n ( x ) = n Y k = j +1 ( W i | k − 1 ) i k , if n > j . F or each I ∈ Q j , we define a sequence Y j,n ( q , I ) := b ndq X i ∈ Λ E ( W q i ) ! − ( n − j ) X J ∈Q n ,J ⊆ I ν j,n ( J ) q . Then, Y j,n ( q , I ) yields the total measure of I for a cascade measure on the interv al I generated by W q . W e next pro vide a v arian t of a classical result of Kahane and Peyri ` ere [10] on the L q -b oundedness of ν n ([0 , 1] d ). Lemma 2.1. Supp ose that 1 ≤ q < p < q max . If 1 < p/q ≤ 2 , for any 1 ≤ j ≤ n and I ∈ Q j , we have E ( Y j,n ( q , I ) p/q ) ≤ b − 2 τ ( p 2 )+ pτ ( q ) q 1 − b − τ ( p )+ pτ ( q ) q . (8) If 2 k < p/q ≤ 2 k +1 for some k ≥ 1 , for any 1 ≤ j ≤ n and I ∈ Q j , then E ( Y j,n ( q , I ) p/q ) ≤  1 − b − τ (2 q )+2 τ ( q )  − 2 k +1 . (9) Pr o of. The pro of is similar to the pro of of Theorem 3 in [7]. Instead of the arith- metic scales k < p/q ≤ k + 1 for k ∈ N used in [7], we pro ceed by up dating E ( Y j,n ( q , I ) p/q ) in dyadic scales, that is 2 k < p/q ≤ 2 k +1 . W e provide the details for the reader’s conv enience. Note that Y j,n +1 ( q , I ) = b − d X i ∈ Λ W q ,i Y j +1 ,n +1 ( q , I i ) where I i , i ∈ Λ are the elements of Q j +1 satisfying I j ⊂ I and ( W q ,i ) i ∈ Λ is an inde- p enden t cop y of W q . F rom now on, W e simply write E ( Y j,n +1 ,q ) := E ( Y j,n +1 ( q , I )) and E ( Y j +1 ,n +1 ,q ) := E ( Y j +1 ,n +1 ( q , I i )), since these expressions are indep enden t of I and I i . Observ e that Y j,n +1 ( q , I ) p q ≤ b − dp q " X i ∈ Λ W p 2 q q ,i Y j +1 ,n +1 ( q , I i ) p 2 q # 2 = b − dp q X i ∈ Λ W p q q ,i Y j +1 ,n +1 ( q , I i ) p q + b − dp q X i 1  = i 2 W p 2 q q ,i 1 Y j +1 ,n +1 ( q , I i 1 ) p 2 q W p 2 q q ,i 2 Y j +1 ,n +1 ( q , I i 2 ) p 2 q . Since Y j +1 ,m ( q , I i ) p/q is a submartingale in m , E ( Y p/q j +1 ,n +1 ,q ) ≤ E ( Y p/q j +1 ,n +2 ,q ) = E ( Y p/q j,n +1 ,q ) . Hence, we ha ve E ( Y p q j,n +1 ,q ) ≤ b − dp q X i ∈ Λ E ( W p q q ,i ) E ( Y p q j,n +1 ,q ) + b − dp q E ( Y p 2 q j +1 ,n +1 ,q ) 2 " X i ∈ Λ E ( W p 2 q q ,i ) # 2 . 6 DONGGEUN R YOU AND VILLE SUOMALA Since b − dp q X i ∈ Λ E ( W p/q q ,i ) = b − τ ( p )+ pτ ( q ) q and b − dp 2 q X i ∈ Λ E ( W p/ 2 q q ,i ) = b − τ ( p 2 )+ pτ ( q ) 2 q , w e obtain that E ( Y p/q j,n +1 ,q ) ≤ b − 2 τ ( p 2 )+ pτ ( q ) q 1 − b − τ ( p )+ pτ ( q ) q E ( Y p/ 2 q j +1 ,n +1 ,q ) 2 . (10) Assume that 2 k < p/q ≤ 2 k +1 for some k ≥ 0. If k = 0, since p/ 2 q < 1, w e hav e E ( Y p/ 2 q j +1 ,n +1 ,q ) ≤ 1. Then, (10) implies (8). If k ≥ 1, w e first note that E ( Y p/ 2 q j +1 ,n +1 ,q ) ≤ E ( Y 2 k j +1 ,n +1 ,q ) , and b − 2 τ ( p 2 )+ pτ ( q ) q ≤ 1 for an y p, q suc h that 1 ≤ q < p/ 2 < q max . Then, we consider p i = 2 i q for 1 ≤ i ≤ k . Using (10) inductiv ely , we obtain that E ( Y p/q j,n +1 ,q ) ≤ min { k,n − j } Y i =1  1 − b − τ (2 k +1 − i q )+2 k +1 − i τ ( q )  − 2 i  1 − b − τ ( p )+ pτ ( q ) q  − 1 . Since τ ( p ) − pτ ( q ) q is an increasing function of p < q max for a fixed q , we get b − τ (2 k +1 − i q )+2 k +1 − i τ ( q ) ≥ b − τ (2 q )+2 τ ( q ) , and b − τ ( p )+ pτ ( q ) q ≥ b − τ (2 q )+2 τ ( q ) . Th us, we hav e established (9). □ Lemma 2.2. F or 1 ≤ p, q ≤ ∞ and 1 ≤ j ≤ n , we let S ( p, q , j, n ) =    X I ∈Q j   X J ∈Q n ,J ⊂ I ν n ( J ) q   p/q    1 /p . Then, we have E ( S ( p, q , j, n )) ≲ b − j e τ ( p ) /p − ( n − j ) e τ ( q ) /q . (11) Note that lim p →∞ e τ ( p ) /p = α min . Thus, in (11) , we adopt the c onvention e τ ( ∞ ) / ∞ = α min if p = ∞ or q = ∞ . Pr o of. F or an y 1 ≤ j ≤ n and I ∈ Q j , we denote S ( q , I , n ) =   X J ∈Q n ,J ⊂ I ν n ( J ) q   1 /q , so that S ( p, q , j, n ) = ( P I ∈Q j S ( q , I , n ) p ) 1 /p . First, w e consider the case when p, q ≤ q max < ∞ . If p ≤ q , the required estimate can b e easily derived by the Minko wski inequality: Since p/q ≤ 1, conditional on ν j , we ha ve E ( S ( q , I , n ) p | ν j ) ≤ E   X J ∈Q n ,J ⊂ I ν n ( J ) q | ν j   p/q = b − ( n − j ) pτ ( q ) /q ν j ( I ) p . FOURIER DIMENSION OF MANDELBROT CASCADES ON PLANAR CUR VES 7 uniformly in ν j . Therefore, E ( S ( p, q , j, n ) | ν j ) ≤   X I ∈Q j E ( S ( q , I , n ) p | ν j )   1 /p = b − ( n − j ) τ ( q ) /q   X I ∈Q j ν j ( I ) p   1 /p . By the la w of total probabilit y , w e thus obtain E ( S ( p, q , j, n )) ≤ b − ( n − j ) τ ( q ) /q E   X I ∈Q j ν j ( I ) p   1 /p = b − ( n − j ) τ ( q ) /q − j τ ( p ) /p . If p, q ≤ q max , recall that e τ ( p ) = τ ( p ) and e τ ( q ) = τ ( q ), resp ectiv ely . Th us, we obtain (11). Next, let us consider the case p, q ≤ q max < ∞ and p > q . W e ha v e S ( q , I , n ) = b − ( n − j ) d X i ∈ Λ E ( W q i ) ! ( n − j ) /q ν j ( I ) Y j,n ( q , I ) 1 /q = b − ( n − j ) τ ( q ) /q ν j ( I ) Y j,n ( q , I ) 1 /q . (12) Therefore, for eac h fixed ν j , we ha ve E ( S ( q , I , n ) p | ν j ) ≤ b − ( n − j ) pτ ( q ) /q ν j ( I ) p E  Y j,n ( q , I ) p/q | ν j  . By Lemma 2.1, we obtain that E ( Y j,n ( q , I ) p/q | ν j ) < ∞ uniformly in j , n , and I . Since Y j,n ( q , I ) and ν j are indep enden t, we obtain that E ( S ( q , I , n ) p | ν j ) ≲ b − ( n − j ) pτ ( q ) /q ν j ( I ) p uniformly in ν j . Rep eating the remaining steps as in the case p, q ≤ q max and p ≤ q , w e obtain the desired estimate (11). If p > q max or q > q max , we get S ( p, q , j, n ) ≤ S (min( p, q max ) , min( q , q max ) , j, n ) . Since e τ ( r ) /r = τ ( q max ) /q max for all r ≥ q max , we obtain (11). Lastly , let us consider the case q max = ∞ . It suffices to consider the case when p = ∞ or q = ∞ . Otherwise, we can rep eat the argumen t abov e. If q < q max and n, j are fixed, then e τ ( p ) /p − → α min and S ( p, q , j, n ) − → S ( ∞ , q, j, n ) as p → ∞ . Also, Lemma 2.1 implies that E ( Y p/q j,n,q ) ≤ C p q when p > 2 q , where C q = (1 − b − τ (2 q )+2 τ ( q ) ) − 2 /q . Therefore, noting (12), the implicit constan t in (11) is uniform in p as p − → ∞ . Th us, (11) for p = q max = ∞ easily follo ws by F atou’s lemma. Next, recall that the implicit constant in (11) equals 1 when p ≤ q and thus (11) for p < ∞ and q = q max = ∞ follows b y the same reasoning as ab o v e. If q max = p = q = ∞ , we use that S ( ∞ , ∞ , j, n ) = S ( ∞ , 1 , n, n ) and (11) easily follo ws. □ F or 1 ≤ p, q ≤ ∞ , w e define ε p,q ,n = 1 n sup { log( S ( p, q , j, n )) + j e τ ( p ) /p + ( n − j ) e τ ( q ) /q : 0 ≤ j ≤ n } . Lemma 2.3. F or fixe d 1 ≤ p, q ≤ ∞ , lim n →∞ ε p,q ,n = 0 almost sur ely on non- extinction. Pr o of. F or an y ε > 0, Lemma 2.2 implies that P  S ( p, q , j, n ) > b − j e τ ( p ) /p − ( n − j ) e τ ( q ) /q + nε  ≤ b − nε , 8 DONGGEUN R YOU AND VILLE SUOMALA and we obtain X n ≥ 0 X 0 ≤ j ≤ n P  S ( p, q , j, n ) > b − j e τ ( p ) /p − ( n − j ) e τ ( q ) /q + nε  < X n ≥ 0 X 0 ≤ j ≤ n b − nε < ∞ . Borel-Can telli lemma implies that, almost surely , there are only finitely many pairs 0 ≤ j ≤ n such that S ( p, q , j, n ) > b − j e τ ( p ) /p − ( n − j ) e τ ( q ) /q + nε . In particular, almost surely , there is n 0 ∈ N such that sup { log( S ( p, q , j, n )) + j e τ ( p ) /p + ( n − j ) e τ ( q ) /q : 0 ≤ j ≤ n } ≤ εn for all n ≥ n 0 . □ Remark 2.4. In the pr o of of the main the or ems, we only use the c ases q = 1 and q = 2 , but the L emmas 2.2 and 2.3 work for al l 1 ≤ p, q ≤ ∞ . 2.3. A concen tration inequality . W e complete this section with our k ey concen- tration inequality , a v ariant of [1, Prop osition C.3]. Lemma 2.5. F or some p > 4 , let Φ( t ) ≲ t − p when t ≥ 1 . L et X 1 , . . . , X N b e indep endent r andom variables with zer o exp e ctation such that P ( X k > t ) ≤ Φ( t ) (13) for al l 1 ≤ k ≤ N . Then, for al l a 1 , a 2 , . . . a N ≥ 0 , M > 1 , and t > 0 , P N X k =1 a k X k > t ! = N Φ( M ) + exp − λt + O ( λ 2 ) N X k =1 a 2 k ! , wher e λ = q log M M max 1 ≤ k ≤ N a k . (14) and q ≤ p/ 2 − 1 . The O -c onstant only dep end on p and Φ . Pr o of. Denote X = P N k =1 a k X k , U k = min { M , X k } , Y k = a k U k , and Y = P N k =1 Y k . Clearly , P ( X > t ) ≤ P ( Y > t ) + P  max 1 ≤ k ≤ N X k > M  . (15) The last term is estimated b y the union b ound P  max 1 ≤ k ≤ N X k > M  ≤ N Φ( M ) . (16) W e pro ceed to estimate P ( Y > t ) using the exponential momen t metho d. Let λ > 0 b e a constant to b e determined later. Using Mark o v’s inequality , we hav e P ( Y > t ) = P (exp( λY ) > exp( λt )) ≤ exp( − λt ) E (exp( λY )) = exp( − λt ) N Y k =1 E (exp( λa k U k )) . (17) F or eac h a > 0, we ha ve E (exp( aU k ))) ≤ 1 + a 2 2 E  U 2 k exp( a max { U k , 0 } )  , (18) see [1, (C.27)]. Let C = C (Φ , p ) suc h that log Φ( t ) ≤ − p log t + C . (19) FOURIER DIMENSION OF MANDELBROT CASCADES ON PLANAR CUR VES 9 Splitting the integral in three parts, using change of v ariables and the tail b ound (13) and (19) yields E  U 2 k exp( a max { U k , 0 } )  = Z U k < 0 U 2 k d P + Z 0 ≤ U k ≤ 1 U 2 k exp( aU k ) d P + Z 1 t  dt ≤ E ( X 2 k ) + exp( a ) + Z M s =1 (2 s + as 2 ) exp( as ) P ( X k > s ) ds ≤ E ( X 2 k ) + exp( a ) + Z M s =1 (2 s + as 2 ) exp ( as + C − p log s ) ds . (20) If a ≤ q log M M , then (since log is conv ex) as ≤ a + q log s , for all 1 ≤ s ≤ M . Whence exp ( C + as − p log s ) ≤ exp ( C + a − (1 + p/ 2) log s ) ≲ s − 1 − p/ 2 . The implicit constant only dep ends on p and Φ, since a ≤ q log M M ≲ q ≲ p and C dep ends on Φ and p . Com bining with (20) and noting that E ( X 2 k ) ≲ 1 b y (13) implies that E  U 2 k exp( a max { U k , 0 } )  ≲ 1 , for all 1 ≤ k ≤ N . Noting (14) and com bining with (17), we ha v e P ( Y > t ) ≤ exp − λt + O ( λ 2 ) N X k =1 a 2 k ! . Com bining with (16), this gives the claim. □ 3. The lower bound of the F ourier dimension W e no w turn to the main no v el feature in this work, the exact v alue of the F ourier dimension for curvilinear cascades. First, w e prov e the lo w er b ound of the F ourier dimension. Theorem 3.1. Almost sur ely, | b µ ( ξ ) | ≲ β | ξ | − β , if β < α min / 2 . This theorem implies that dim F ( µ ) ≥ α min . W e will consider differen t estimates according to the size of | ξ | . Lemma 3.2. If β < e τ (2) , then | [ µ n +1 ( ξ ) − c µ n ( ξ ) | ≲ β b − nβ / 2 for al l n and al l | ξ | ≤ b n , wher e the implicit c onstant is (r andom and) indep endent of n , ξ . W e defer the pro of of Lemma 3.2 to the app endix, see Lemma 6.3 (and Remark 6.4), where a sligh tly more general version is obtained. W e note that for Theorem 3.1, w e only need the bound for β < α min . The full p ow er of the lemma will be used for the spherical av erages in Section 5. W e will use the following, which is an immediate consequence of our curv ature assumption and the V an der Corput lemm a (see e.g. [17, Theorem 14.2]). Lemma 3.3. L et I ⊂ [0 , 1] . Then      Z γ ( I ) exp ( − 2 π ix · ξ ) dx      ≲ | ξ | − 1 / 2 . 10 DONGGEUN R YOU AND VILLE SUOMALA If 1 ≤ j ≤ n and b − j | ξ | ≤ | γ ′ ( t ) · ξ | for al l t ∈ I , then      Z γ ( I ) exp ( − 2 π ix · ξ ) dx      ≲ b j / | ξ | , W e will use the follo wing notation in the pro ofs of Lemmas 3.4– 3.6, and 5.3. Giv en ξ , let I 1 = I 1 ( ξ ) consist of in terv als I ∈ D n suc h that b − 1 | ξ | ≤ min {| γ ′ ( t ) · ξ | : γ ( t ) ∈ I } . F or 2 ≤ j < n , let I j = I j ( ξ ) consist of those in terv als I ∈ D n suc h that b − j | ξ | ≤ min {| γ ′ ( t ) · ξ | : γ ( t ) ∈ I } < b − j +1 | ξ | and let I n = I n ( ξ ) = D n \ ∪ n − 1 j =1 I j . Note that ∪ I ∈I j I is con tained in a union of O (1)-man y arcs of length ∼ b − j . Let us denote the union of these arcs b y S j = S j ( ξ ). No w, we can prov e the estimates for c µ n ( ξ ) in different scales of ξ . Throughout this section, let us denote ε n = ε ∞ , 1 ,n for notational con v enience. Lemma 3.4. If β < α min , then c µ n ( ξ ) ≲ β | ξ | ( ε n − β ) / 2 for al l | ξ | ≥ b 2 n , wher e the implicit c onstant is (deterministic and) indep endent of n and ξ . Pr o of. Consider ξ ∈ R 2 , | ξ | ≥ b 2 n . Using Lemma 3.3 for each I ∈ D n and summing o ver all interv als I ∈ I j yields that if 1 ≤ j < n , then       X I ∈I j Z I exp ( − 2 π ix · ξ ) dµ n ( x )       ≲ b n + j µ n ( S j ) / | ξ | ≲ b n + j (1 − α min )+ nε n / | ξ | , (21) and      X I ∈I n Z I exp ( − 2 π ix · ξ ) dµ n ( x )      ≲ b n µ n ( S n ) | ξ | − 1 / 2 ≲ b n (1 − α min + ε n ) | ξ | − 1 / 2 . Note that the sum of (21) ov er 1 ≤ j ≤ n is ≲ nb 2 − α min + ε n | ξ | − 1 and w e can drop n if α min < 1. This yields | c µ n ( ξ ) | ≲  b n (1 − α min + ε n ) | ξ | − 1 / 2 + nb n (2 − α min + ε n ) | ξ | − 1  ≲ β | ξ | ( ε n − β ) / 2 , pro ving the lemma. □ In the in termediate scales, w e com bine the concen tration inequalit y (Lemma 2.5) with v an der Corput’s lemma. Some elemen tary but technical parts of the pro of are identical to those in the pro of of Lemma 6.3, and they are omitted here. F or t > 0, we let A t denote the union of all I ∈ D n whic h in tersect the closed t -neigh b ourhoo d of a set A ⊂ R d . Lemma 3.5. F or any δ > 0 , almost sur ely,   [ µ n +1 ( ξ ) − c µ n ( ξ )   ≲ δ b 2 nδ   X 1 ≤ ℓ ≤ k b 2( ℓ − k ) X i ∈ Λ n ,D i ⊆ ( S ℓ ( ξ )) b 1 − n µ n ( D i ) 2   1 / 2 + b − n , for al l 1 ≤ k ≤ n and b n + k − 1 ≤ | ξ | ≤ b n + k . Her e, the implicit c onstant is r andom, but indep endent of n, k and ξ . Pr o of. F or 1 ≤ ℓ ≤ k and for each D i ∈ I ℓ , let a i ,ξ = b ℓ − k µ n ( D i ) and X i ,ξ = b n + k − ℓ X j ∈ Λ (( W i ) j − 1) Z D i ,j exp( − 2 π ix · ξ ) dx FOURIER DIMENSION OF MANDELBROT CASCADES ON PLANAR CUR VES 11 where ( W i ) j denotes an indep endent copy of W and D i ,j = γ ( Q i ,j ) where Q i ,j = x ( i ,j ) + [0 , b − ( n +1) ) with ( i , j ) = ( i 1 , i 2 , · · · , i n , j ) for each i ∈ Λ n and j ∈ Λ. Then, w e can write [ µ n +1 ( ξ ) − c µ n ( ξ ) = X i ∈ Λ n a i ,ξ X i ,ξ . Lemma 3.3 implies that b n + k − ℓ Z D i ,j exp( − 2 π ix · ξ ) dx ≲ 1 uniformly in D i ,j , n, k and ℓ . Also, (6) implies that P  max j ∈ Λ W j > t + 1  ≲ p t − p for all 0 < p < ∞ . Thus, P ( X i ,ξ > t ) ≲ p t − p for all X i ,ξ when t > 1. F or eac h n and k , denote S n,k,ξ = X 0 ≤ ℓ ≤ k b 2( ℓ − k ) X D i ⊆S ℓ ( ξ ) µ n ( D i ) 2 . W e ma y use Lemma 2.5 with N = b n +1 , q = 1, t = b 2 nδ S 1 / 2 n,k,ξ and λ = t − 1 b nδ and conclude (see the pro of of Lemma 6.3) that P ( | [ µ n +1 ( ξ ) − c µ n ( ξ ) | ≥ b 2 nδ S 1 / 2 n,k,ξ ) ≤ 2 b n +1 Φ( M ) + 2 exp( − b nδ + O ( b − 2 nδ )) where log( M ) M = λ max { a i ,ξ : i ∈ Λ n } ≲ b − nδ . Cho osing p > 7 /δ , this implies that P  | [ µ n +1 ( ξ ) − c µ n ( ξ ) | ≥ b 2 nδ S 1 / 2 n,k,ξ for some ξ ∈ b − n Z 2 , b n ≤ | ξ | ≤ b 2 n  ≲ δ b − cn for some c > 0. The Borel-Can telli lemma implies that almost surely | [ µ n +1 ( ξ ) − c µ n ( ξ ) | ≲ δ b 2 nδ S 1 / 2 n,k,ξ , (22) for all ξ ∈ b − n Z 2 . Thus, for any b n + k − 1 ≤ | ξ | ≤ b n + k , there exists ξ ′ ∈ b − n Z suc h that | ξ − ξ ′ | ∞ ≤ b − n and (22) holds for ξ ′ . F or Γ = γ ([0 , 1]), µ n (Γ) is a martingale with E ( µ n (Γ)) = 1. Hence, sup n µ n (Γ) < ∞ almost surely . This implies that c µ n ( ξ ) is Lipschitz with a Lipschitz constant indep enden t of n and using this, (22), and the fact S ℓ ( ξ ′ ) ⊂ ( S ℓ ( ξ )) b 1 − n , if | ξ − ξ ′ | ∞ ≤ b − n , we obtain the desired conclusion for all b n + k − 1 ≤ | ξ | ≤ b n + k . □ Lemma 3.6. L et β < α min . Then, almost sur ely, | [ µ n +1 ( ξ ) − c µ n ( ξ ) | ≲ β | ξ | − β / 2 for al l n and al l b n ≤ | ξ | ≤ b 2 n , wher e the implicit c onstant is r andom, but inde- p endent of n and ξ . Pr o of. Let 0 < δ < ( α min − β ) / 4, 1 ≤ k ≤ n and b n + k − 1 ≤ | ξ | ≤ b n + k . Recall that S ℓ = S ℓ ( ξ ) is a union of O (1)-man y arcs of length ∼ b − ℓ . Th us, we hav e X i ∈ Λ n ,D i ⊆ ( S ℓ ( ξ )) b 1 − n µ n ( D i ) 2 ≤ b − nα min + nε n X i ∈ Λ n ,D i ⊆ ( S ℓ ( ξ )) b 1 − n µ n ( D i ) ≲ b − ( n + ℓ ) α min +2 nε n . 12 DONGGEUN R YOU AND VILLE SUOMALA Therefore, X 0 ≤ ℓ ≤ k b 2( ℓ − k ) X i ∈ Λ n ,D i ⊆ ( S ℓ ( ξ )) b 1 − n µ n ( D i ) 2 ≲ b 2 nε n − ( n + k ) α min ∼ b 2 nε n | ξ | − α min . (23) Since | ξ | ≤ b 2 n and α min ≤ 1, w e also ha v e b − n ≲ | ξ | − α min / 2 . Thus, com bining (23) and Lemma 3.5, it follows that | [ µ n +1 ( ξ ) − c µ n ( ξ ) | ≲ | ξ | − α min / 2 b n (2 δ + ε n ) . This implies the claim since Lemma 2.3 applied with p = ∞ and q = 1 yields ε n − → 0 as n → ∞ , almost surely . □ No w, we are ready to prov e Theorem 3.1. Pr o of of The or em 3.1. Let 0 < β < β ′ < α min . Using Lemmas 3.2 and 3.6, it follo ws that for some random constant K > 0, | [ µ n +1 ( ξ ) − c µ n ( ξ ) | ≤ K b − β n/ 2 for all n ∈ N and all | ξ | ≤ b n and, moreov er, that | [ µ n +1 ( ξ ) − c µ n ( ξ ) | ≤ K | ξ | − β ′ / 2 if b n ≤ | ξ | ≤ b 2 n . Consider ξ with b 2 j ≤ | ξ | ≤ b 2 j +2 , j ∈ N and let m ≥ 2 j . Applying Lemma 3.4, w e get | c µ j ( ξ ) | + m X n = j | [ µ n +1 ( ξ ) − c µ n ( ξ ) | = | c µ j ( ξ ) | + 2 j X n = j | [ µ n +1 ( ξ ) − c µ n ( ξ ) | + m X n =2 j | [ µ n +1 ( ξ ) − c µ n ( ξ ) | ≲ β | ξ | ( ε j − β ) / 2 + | ξ | − β / 2 + | ξ | − β / 2 , where we ha v e used that j ≲ log | ξ | ≲ | ξ | ( β ′ − β ) / 2 . Noting that the implicit constan t is independent of ξ , that j → ∞ as | ξ | → ∞ , and that ε j → 0 as j → ∞ (by Lemma 2.3 applied with p = ∞ and q = 1), w e get | b µ ( ξ ) | = lim m →∞ | c µ m ( ξ ) | ≲ β | ξ | − β / 2 . □ 4. The upper bound of the F ourier dimension The upp er b ound on the F ourier dimension, dim F µ ≤ α min , is a general fact v alid for all measures supported on sufficien tly regular curv es. The follo wing lemma is likely known, but we provide a pro of since we hav e not found the result in the literature. F or eac h θ ∈ S d − 1 , we let P θ : R d → R denote the pro jection P θ ( x ) = x · θ , and let η θ b e the image measure defined as η θ ( B ) = η ( P − 1 θ ( B )) for B ⊂ R . Lemma 4.1. L et η b e a finite Bor el me asur e supp orte d on a C 2 -curve Γ ⊂ R 2 define d by γ ( t ) : [0 , 1] → R 2 with det( γ ′ ( t ) , γ ′′ ( t ))  = 0 . F or al l x ∈ spt η , dim F η ≤ dim( η , x ) . The estimate dim F µ ≤ α min is an immediate corollary to Lemma 4.1. Pr o of. Let x ′ ∈ spt η . W e will show that dim 2 η θ ≤ dim( η , x ′ ), where θ ∈ S 1 is the unit normal of Γ at x ′ . Using the iden tit y b η θ ( ξ ) = b η ( ξ θ ) , it then follo ws that dim F η ≤ dim F η θ ≤ dim 2 η θ ≤ dim( η , x ′ ). Let t > s > dim( η , x ′ ). It suffices to show that dim 2 η θ ≤ t . T o that end, we recall that dim 2 η θ = sup  0 ≤ h < 1 : Z Z | x − y | − h dη θ ( x ) dη θ ( y ) < ∞  , FOURIER DIMENSION OF MANDELBROT CASCADES ON PLANAR CUR VES 13 see e.g. [8, Proposition 2.1]. Since det( γ ′ ( t ) , γ ′′ ( t )) > 0 and θ is the unit normal of Γ at x ′ , there is 1 ≤ C < ∞ , such that Γ ∩ B ( x ′ , r 1 / 2 ) ⊂ P − 1 θ ( B ( x, C r )) for all 0 < r < 1, and for all x ∈ B ( P θ ( x ′ ) , r )) ⊂ R . Now Z Z | x − y | − t dη θ ( x ) dη θ ( y ) ≳ r − t Z η θ ( B ( x, C r )) dη θ ( x ) , ≥ r − t Z B ( P θ ( x ′ ) ,r ) η ( B ( x ′ , r 1 / 2 )) dη θ ( x ) ≳ r − t η  B ( x ′ , C − 1 / 2 r 1 / 2 )  2 for all 0 < r < 1. No w, there are arbitrarily small v alues 0 < r < 1, suc h that η ( B ( x ′ , C − 1 / 2 r 1 / 2 )) > r s/ 2 , and for these v alues of r , w e thus ha ve Z Z | x − y | − t dη θ ( x ) dη θ ( y ) ≳ r s − t , implying that dim 2 η θ ≤ t . □ 5. Deca y of the spherical a verage Recall the definition of the spherical av erage σ p from (2). Theorem 5.1. L et β < min { e τ (2) , (1 + e τ ( p )) /p } . F or 1 ≤ p ≤ ∞ , almost sur ely we have σ p ( µ )( r ) ≲ β r − β / 2 . Remark 5.2. a) If 1 ≤ p ≤ 2 , then min { e τ (2) , (1 + e τ ( p )) /p } = e τ (2) . (24) Inde e d, sinc e (1 + e τ ( p )) /p is a slop e b etwe en ( p, e τ ( p )) and (0 , − 1) , and e τ ( p ) is a c onc ave function of p , (1 + e τ (2)) / 2 ≤ (1 + e τ ( p )) /p if 1 ≤ p ≤ 2 . Thus, (24) fol lows fr om the fact that e τ (2) ≤ 1 . b) The b or derline c ase p = ∞ is e quivalent to The or em 3.1 and thus we assume in the pr o of that p < ∞ . We note that with natur al L ∞ -interpr etations, the pr o of b elow would also c over the p = ∞ c ase (and this would essential ly r ep e at the pr o of of The or em 3.1). Lemma 5.3. If β < (1 + e τ ( p )) /p , then σ p ( µ n )( r ) ≲ β r ( ε p, 1 ,n − β ) / 2 , for al l r ≥ b 2 n , wher e the implicit c onstant is (deterministic and) indep endent of n . Pr o of. Recall the notations I j ( ξ ), S j ( ξ ) from Section 3. If θ ∈ S 1 and I ∈ I j ( θ ) for 1 ≤ j ≤ n − 1, we use Lemma 3.3 and obtain that     Z I exp( − 2 π irθ · x ) dµ n ( x )     ≲ b n + j µ n ( I ) r − 1 , and if I ∈ I n ( θ ), then     Z I exp( − 2 π irθ · x ) dµ n ( x )     ≲ b n µ n ( I ) r − 1 / 2 . Therefore, if 1 ≤ j ≤ n − 1, then we ha v e       X I ∈I j ( θ ) Z I exp( − 2 π irθ · x ) dµ n ( x )       L p ( dσ ) ≲ b n + j r − 1 ∥ µ n ( S j ( θ )) ∥ L p ( dσ ) . 14 DONGGEUN R YOU AND VILLE SUOMALA Recall that S j ( θ ) is contained in a union of O (1)-man y arcs of length b − j , and that S 1 ( I ) := { θ ∈ S 1 | I ∩ S j ( θ )  = ∅ } is an arc of length ∼ b − j for I ∈ D j . Th us, w e obtain ∥ µ n ( S j ( θ )) ∥ p L p ( dσ ) ≲ Z       X I ∈D j ,I ∩S j ( θ )  = ∅ µ n ( I )       p dσ ( θ ) ≲ X I ∈D j µ n ( I ) p σ ( S 1 ( I )) ≲ b − j X I ∈D j µ n ( I ) p . Putting things together, we ha v e       X I ∈I j ( θ ) Z I exp( − 2 π irθ · x ) dµ n ( x )       L p ( dσ ) ≲ b n + j r − 1   b − j X I ∈D j µ n ( I ) p   1 /p ≲ ( b n (1+ ε p, 1 ,n )+ j (1 − 1 /p − e τ ( p ) /p ) ) r − 1 , where we used the fact that e τ (1) = 0. Similarly , if n = j , then       X I ∈I n ( θ ) Z I exp( − 2 π irθ · x ) dµ n ( x )       L p ( dσ ) ≲ b n r − 1 / 2 ∥ µ n ( S j ( θ )) ∥ L p ( dσ ) ≲ b n r − 1 / 2 b − n X I ∈D n µ n ( I ) p ! 1 /p ≲ ( b n (1+ ε p, 1 ,n − 1 /p − e τ ( p ) /p ) ) r − 1 / 2 . W e sum the ab o v e estimates ov er 0 ≤ j ≤ n and use that r ≥ b 2 n , e τ ( p ) ≤ p − 1. This implies σ p ( µ n )( r ) ≲ ( b n (1+ ε p, 1 ,n − 1 /p − e τ ( p ) /p ) )( nb n r − 1 + r − 1 / 2 ) ≲ β r ( ε p, 1 ,n − β ) / 2 , as required. □ Lemma 5.4. L et b n ≤ r ≤ b 2 n . F or any δ > 0 , almost sur ely we have σ p ( µ n +1 − µ n )( r ) ≲ b − n ( e τ (2) − (1+ e τ ( p )) /p − 3 δ ) r e τ (2) / 2 − (1+ e τ ( p )) /p . Pr o of. Let δ > 0. Let 1 ≤ k ≤ n such that b n + k − 1 ≤ r ≤ b n + k and let ξ = r θ , where θ ∈ S 1 . Note that S ℓ ( ξ ) = S ℓ ( θ ). Lemma 3.5 implies that almost surely we ha ve σ p ( µ n +1 − µ n )( r ) ≲ b 2 nδ X 1 ≤ ℓ ≤ k b ℓ − k       X i ∈ Λ n ,D i ⊆ ( S ℓ ( θ )) b 1 − n µ n ( D i ) 2       L p ( dσ ) + b − n ≲ b 2 nδ X 1 ≤ ℓ ≤ k b ℓ − k    b − ℓ X I ∈D ℓ       X i ∈ Λ n ,D i ⊆ ( S ℓ ( θ )) b 1 − n µ n ( D i ) 2       p/ 2    1 /p + b − n ≲ b 2 nδ X 1 ≤ ℓ ≤ k b ℓ − k ( b − ℓ (1+ e τ ( p )) /p − ( n − ℓ ) e τ (2) / 2 ) b nε p, 2 ,n + b − n , FOURIER DIMENSION OF MANDELBROT CASCADES ON PLANAR CUR VES 15 where, in the second inequality , w e hav e again used that eac h I ∈ D ℓ b elongs to ( S ℓ ( θ )) b 1 − n for θ in an interv al of length ∼ b − ℓ . F or sufficiently large n , Lemma 2.3 implies that ε p, 2 ,n < δ . Since r ∼ b n + k , and 1 − (1 − e τ ( p )) /p + e τ (2) / 2 > 0, the desired result follo ws by summing ov er ℓ . □ No w, we can prov e Theorem 5.1. Pr o of of The or em 5.1. Consider r with b 2 j ≤ r ≤ b 2 j +2 , j ∈ N and let m ≥ 2 j . Lemma 3.2 implies that if β < e τ (2), then almost surely , m X n =2 j σ p ( µ n +1 − µ n )( r ) = m X n =2 j b − nβ / 2 ≲ r − β / 2 . where the implicit constant is indep enden t of n and k . Also, Lemmas 2.3 and 5.3 imply that if β < (1 + e τ ( p )) /p , then σ p ( µ j )( r ) ≲ r − β / 2 almost surely . If j ≤ n ≤ 2 j , then Lemma 5.4 imply that almost surely w e hav e 2 j X n = j σ p ( µ n +1 − µ n )( r ) ≲ 2 j X n = j b − n ( e τ (2) − (1+ e τ ( p )) /p − 3 δ ) r e τ (2) / 2 − (1+ e τ ( p )) /p ≲ r − e τ (2) / 2+3 δ + r − (1+ e τ ( p )) / 2 p +3 δ / 2 . Com bining the estimates ab o v e, we obtain that for β < min { e τ (2) , (1 + e τ ( p )) /p } , σ p ( µ m )( r ) ≤ σ p ( µ j )( r ) + 2 j X n = j σ p ( µ n +1 − µ n )( r ) + m X n =2 j σ p ( µ n +1 − µ n )( r ) ≲ r − β / 2 , completing the proof. □ 6. Appendix Let ν be the Mandelbrot cascade on [0 , 1] d corresp onding to a random v ariable W satisfying (5) and (6). The following theorem generalizes the main result of [3]. The result, as stated here, is a special case of more general results obtained recen tly in [4, 12, 13]. Compared to these pap ers, our approac h b elo w provides a simple pro of under the fairly general moment condition (6). Theorem 6.1. dim F ν = min { 2 , dim 2 ν } almost sur ely on non-extinction. Recall that dim F η ≤ dim 2 η holds for any finite Borel measure η . The upp er b ound dim F ν ≤ 2 is prov ed in [3, Theorem 3.2] in the case of iid W i , but the pro of extends to general cascades without an y difficulty . It thus remains to prov e the lo wer b ound dim F ν ≥ min { 2 , dim 2 ν } . W e first recall the follo wing lemma, whic h rev eals, in particular, that e τ (2) is almost surely the same as dim 2 ν . Lemma 6.2. Almost sur ely on non-extinction, dim 2 ( ν ) = lim n →∞ log P I ∈Q n ν n ( I ) 2 − n = e τ (2) . Pr o of. F or all q > 1, we define auxiliary random v ariables W q = b d ( W q j ) j ∈ Λ P j ∈ Λ E ( W q j ) . Note that eac h W q satisfies (3). Denote Y n ( q ) := b ndq   X j ∈ Λ E ( W q j )   − n X I ∈Q n ν n ( I ) q . 16 DONGGEUN R YOU AND VILLE SUOMALA Then Y n is the total measure at generation n of the cascade corresp onding to W q . If W q is sub critical, Y n ( q ) con verges to a nonzero limit almost surely , on non-extinction, and whence lim n →∞ log P I ∈Q n ν n ( I ) q − n = τ ( q ) . (25) After this, the pro of pro ceeds verbatim with [3, Lemma 2.2]. □ The heart of the pro of of Theorem 6.1 is the following application of Lemma 2.5. Lemma 6.3. L et β < e τ (2) . Then, almost sur ely, | [ ν n +1 ( ξ ) − c ν n ( ξ ) | ≲ β b − nβ / 2 (26) holds for al l n ∈ N and al l ξ ∈ R d , | ξ | ∞ ≤ b n +1 . Her e, the implicit c onstant is r andom, but indep endent of n and ξ . Pr o of. Let δ > 0. Given ξ and n , we ma y write [ ν n +1 ( ξ ) − c ν n ( ξ ) = X i ∈ Λ n ν n ( Q i ) X i ,ξ , where X i ,ξ = b nd X j ∈ Λ (( W i ) j − 1) Z Q i ,j exp( − 2 π ix · ξ ) dx , and W i is an indep endent copy of W and Q i ,j = x ( i ,j ) + [0 , b − ( n +1) ) d with ( i , j ) = ( i 1 , i 2 , · · · , i n , j ) for eac h i ∈ Λ n and j ∈ Λ. W e note that, conditional on F n (the sigma-algebra generated by W i , i ∈ ∪ n k =0 Λ k ), the random v ariables X i ,ξ are indep enden t, hav e zero mean, and satisfy P ( | r eX i ,ξ | > t ) , P ( | imX i ,ξ | > t ) ≤ Φ( t ) := P  max j ∈ Λ W j > t + 1  . Note that (6) implies Φ( t ) ≲ p t − p for all 0 < p < ∞ . F or eac h n ∈ N , denote S n = P Q i ∈Q n ν 2 n ( Q i ). W e may then use the Lemma 2.5 for the real and imaginary parts of X i ,ξ with the choice N = b nd , q = 1, t = b 2 nδ S 1 / 2 n and λ = t − 1 b nδ . Then, the lemma yields P    [ ν n +1 ( ξ ) − c ν n ( ξ )   > b 2 nδ S 1 / 2 n  = 2 b nd Φ( M ) + 2 exp( − b nδ + O ( b − 2 nδ )) , where M log M = 1 λ max i ∈ Λ n ν n ( Q ) ≥ b nδ and whence w e choose M ≥ b nδ . Com bining with a union b ound, P    [ ν n +1 ( ξ ) − c ν n ( ξ )   > b 2 nδ S 1 / 2 n for some ξ ∈ b − nd/ 2 Z d , | ξ | ∞ ≤ b n +1  ≲ b n ( d 2 / 2+ d )  b nd Φ( b nδ ) + exp( − b nδ )  . Using that Φ( t ) ≲ t − p for all p and choosing p > ( d 2 / 2 + 2 d ) /δ , w e hav e P    [ ν n +1 ( ξ ) − c ν n ( ξ )   > b 2 nδ S 1 / 2 n for some ξ ∈ b − nd/ 2 Z d , | ξ | ∞ ≤ b n +1  ≲ δ b − nc , for some c = c δ > 0. Applying the Borel-Can telli lemma, it follo ws that, almost surely , | [ ν n +1 ( ξ ) − c ν n ( ξ ) | ≤ b 2 nδ S 1 / 2 n for all ξ ∈ b − nd/ 2 Z d , | ξ | ∞ ≤ b n +1 and for all except finitely man y n . FOURIER DIMENSION OF MANDELBROT CASCADES ON PLANAR CUR VES 17 Since ν n ([0 , 1] d ) is a martingale with E ( ν n ([0 , 1] d )) = 1, w e hav e sup n ν n ([0 , 1] d ) < ∞ almost surely . Therefore, c ν n ( ξ ) is Lipschitz with a Lipschitz constan t indep en- den t of n . Thus, we observ e that, almost surely , | [ ν n +1 ( ξ ) − c ν n ( ξ ) | ≲ b 2 nδ S 1 / 2 n + b − nd/ 2 for all | ξ | ∞ ≤ b n +1 , and for all except finitely many n . Com bining with Lemma 6.2 and the fact that e τ (2) ≤ d , this completes the pro of. □ Remark 6.4. We note that the ab ove pr o of works, verb atim, for the curviline ar c asc ades µ n . In this c ase, d = 1 and the inte gr als in the definition of X i ,j ar e over γ ( Q i ,j ) . Thus, we have also verifie d the L emma 3.2. Pr o of of The or em 6.1. Once w e ha v e Lemma 6.3 at our disp osal, the proof is iden- tical to the pro of of [3, Theorem 3.1] (see also the pro of of [20, Theorem 14.1]). W e pro vide the details for the conv enience of the reader. Let β < min { 2 , e τ (2) } . Almost surely , there is a finite implicit constan t such that the claim of Lemma 6.3 holds. Conditional on this, we compute as follo ws (once β and the implicit O (1)-constan t of Lemma 6.3 are fixed, the pro of do es not con tain any probabilistic elements): If | ξ | ≤ b n +1 , we ma y use (26). If | ξ | > b n +1 , w e write ξ = b n +1 q + ξ ′ where q ∈ Z d and | ξ ′ | < b n +1 . Using c hange of v ariables in each co ordinate (this is applied to compute d 1 Q i ( ξ ) for i ∈ Λ i ) gives the identit y [ ν n +1 ( ξ ) − c ν n ( ξ ) =  [ ν n +1 ( ξ ′ ) − c ν n ( ξ ′ )  Y 1 ≤ j ≤ d , ξ j  =0 | ξ ′ j | | ξ j | . Eac h term in the ab ov e pro duct is ≤ 1 and the smallest term is ≲ b n +1 / | ξ | ∞ . Com bining this with (26) (applied to ξ ′ ), this giv es | [ ν n +1 ( ξ ) − c ν n ( ξ ) | ≤ b n +1 | ξ | ∞   [ ν n +1 ( ξ ′ ) − c ν n ( ξ ′ )   ≤ b n (1 − β / 2) | ξ | ∞ . (27) W e may no w finish the pro of as follows. If ξ ∈ R d , | ξ | ∞ > b , let n ξ ∈ N so that b n ξ < | ξ | ∞ ≤ b n ξ +1 . Using (27) for n ≤ n ξ and (26) for n > n ξ and telescoping, w e ha ve | c ν m ( ξ ) − b ν 0 ( ξ ) | ≤ n ξ X n =0 | [ ν n +1 ( ξ ) − c ν n ( ξ ) | + m − 1 X n = n ξ +1 | [ ν n +1 ( ξ ) − c ν n ( ξ ) | ≲ | ξ | − 1 n ξ X n =0 b n (1 − β / 2) + m − 1 X n = n ξ +1 b − nβ / 2 ≲ | ξ | − β / 2 , for all m ≥ n ξ . Since also b ν 0 ( ξ ) ≲ | ξ | − 1 ≲ β | ξ | − β / 2 , we get b ν ( ξ ) = lim m →∞ c ν m ( ξ ) ≲ β | ξ | − β / 2 + | ξ | − 1 ≲ | ξ | − β / 2 , where the implicit constan t is independent of ξ . Since β < min { 2 , e τ (2) } is arbitrary , together with Lemma 6.2, this implies that dim F ν ≥ min { 2 , dim 2 ν } almost surely on non-extinction. □ References [1] S. Armstrong and T. Kuusi. Renormalization group and elliptic homogenization in high con- trast. Invent. Math. , 242(3):895–1086, 2025. [2] J. Barral. Contin uity of the m ultifractal spectrum of a random statistically self-similar mea- sure. J. Theor et. Pr obab. , 13(4):1027–1060, 2000. 18 DONGGEUN R YOU AND VILLE SUOMALA [3] C. Chen, B. Li, and V. Suomala. F ourier dimension of Mandelbrot multiplicativ e cascades. Comm. Math. Phys. , 406(8):Paper No. 182, 15, 2025. [4] X. Chen, Y. Han, Y. Qiu, and Z. W ang. Harmonic analysis of Mandelbrot cascades–in the context of vector-v alued martingales. Preprint, av ailable at [5] K. F alconer and X. Jin. Exact dimensionality and pro jection prop erties of Gaussian m ulti- plicative chaos measures. T r ans. Amer. Math. So c. , 372(4):2921–2957, 2019. [6] C. Garban and V. V argas. Harmonic analysis of Gaussian multiplicative chaos on the c ircle. Preprint, av ailable at [7] Y. Heurteaux. An introduction to Mandelbrot cascades. In New tr ends in applied harmonic analysis , Appl. Numer. Harmon. Anal., pages 67–105. Birkh¨ auser/Springer, Cham, 2016. [8] B. R. Hun t and V. Y. Kaloshin. How pro jections affect the dimension sp ectrum of fractal measures. Nonline arity , 10(5):1031–1046, 1997. [9] J.-P . Kahane. F ractals and random measures. Bul l. Sci. Math. , 117(1):153–159, 1993. [10] J.-P . Kahane and J. Peyri ` ere. Sur certaines martingales de Benoit Mandelbrot. A dvanc es in Math. , 22(2):131–145, 1976. [11] R. Kaufman. On the theorem of Jarn ´ ık and Besicovitc h. Acta Arith. , 39(3):265–267, 1981. [12] Z. Lin, Y. Qiu, and M. T an. Harmonic analysis of multiplicative chaos Part I: the pro of of Garban-Vargas conjecture for 1D GMC. Preprint, av ailable at arXiv:2411:13923. [13] Z. Lin, Y. Qiu, and M. T an. Harmonic analysis of multiplicativ e chaos Part I I: a unified approach to Fourier dimensions. Preprint, av ailable at [14] R. Lyons. Seven ty y ears of Ra jchman measures. In Pr o c e e dings of the Confer ence in Honor of Jean-Pierr e Kahane (Orsay, 1993) , pages 363–377, 1995. [15] B. Mandelbrot. Intermitten t turbulence in self similar cascades: divergence of high moments and dimension of carrier. J. Fluid Me ch. , 62:331–333, 1974. [16] B. Mandelbrot. Intermitten t turbulence and fractal dimension: kurtosis and the sp ectral exponent 5 / 3 + B . In T urbulence and Navier-Stokes e quations (Pro c. Conf., Univ. Paris- Sud, Orsay, 1975) , volume V ol. 565 of L e ctur e Notes in Math. , pages 121–145. Springer, Berlin-New Y ork, 1976. [17] P . Mattila. F ourier analysis and Hausdorff dimension , v olume 150 of Cambridge Studies in A dvanc e d Mathematics . Cambridge Universit y Press, Cambridge, 2015. [18] G. M. Molchan. Scaling exp onen ts and multifractal dimensions for indep endent random cas- cades. Comm. Math. Phys. , 179(3):681–702, 1996. [19] D. Ryou. Near-optimal restriction estimates for Cantor sets on the parabola. Int. Math. R es. Not. IMRN , (6):5050–5099, 2024. [20] P . Shmerkin and V. Suomala. Spatially indep enden t martingales, intersections, and applica- tions. Mem. Amer. Math. So c. , 251(1195):v+102, 2018. DR: Dep ar tment of Ma thema tics, Indiana University, Bloomington, IN, USA. Email address : dryou@iu.edu VS: Ma thema tical Sciences, P.O.Box 8000, FI-90014, University of Oulu, Finland Email address : ville.suomala@oulu.fi