Convergence over fractals for the Schrödinger equation
aa r X i v : . [ m a t h . A P ] J a n Convergence over fractals for the Schrödingerequation
Lucà, R. and Ponce-Vanegas, F.
Abstract
We consider a fractal refinement of the Carleson problem for theSchrödinger equation, that is to identify the minimal regularity neededby the solutions to converge pointwise to their initial data almost ev-erywhere with respect to the α -Hausdorff measure ( α -a.e.). We extendto the fractal setting ( α < n ) a recent counterexample of Bourgain [5],which is sharp in the Lebesque measure setting ( α = n ). In doingso we recover the necessary condition from [23] for pointwise conver-gence α -a.e. and we extend it to the range n/ < α ≤ (3 n + 1) / . A classic question related to solutions to the linear Schrödinger equation(here ~ = 1 / (2 π ) ) ∂ t u = i ~ uu ( x,
0) = f ( x ) ∈ H s ( R n ) , is: what is the minimal regularity the initial datum must have so that thesolution u converge almost everywhere (a.e.) to f ? More precisely, which isthe smallest s ≥ such that lim t → u ( x, t ) = f ( x ) , for a.e. x ∈ R n and for all f ∈ H s ( R n ) . (1)This problem was introduced by Carleson in [8], where he proved the validityof (1) for s ≥ / in dimension n = 1 . Soon later Dahlberg and Kenig [10]proved this to be sharp. The considerably harder higher dimensional problem1as subsequently studied by many authors [9, 6, 30, 34, 3, 26, 32, 33, 31, 22,4, 25, 11, 24, 14].Recently, the problem has been settled, up to the endpoint, thanks to thecontributions of Bourgain [5] (see [27] for a nice detailed exposition), whoproved the necessity of s ≥ n n +1) , and of Du–Guth–Li [13] and of Du–Zhang[15], who proved the sufficiency of s > n n +1) in dimensions n = 2 and n ≥ , respectively. We mention that, besides Bourgain’s counterexample, thenecessity of s ≥ n n +1) can be proved also by different counterexamples [23].In this paper we consider a fractal refinement of the Carleson problem.Given α ∈ (0 , n ] , the goal is to identify the smallest ≤ s ≤ n/ such that lim t → u ( x, t ) = f ( x ) , for α -a.e. x ∈ R n and for all f ∈ H s ( R n ) , (2)where α -a.e. means almost everywhere with respect to the α -dimensionalHausdorff measure.This fractal refinement of the Carleson problem was introduced in [29].In [2], the authors gave a complete solutions for α ∈ [0 , n/ , proving that s > ( n − α ) / is necessary and sufficient for (2) to hold. The necessity of thiscondition depends on the Sobolev space framework, since for smaller s thereexist initial data in H s ( R n ) that are not well defined on sets of dimension α ; see [35]. On the other hand, for s > ( n − α ) / one can make sense ofthe initial data and of the relative solution α -a.e.; we refer to the proof ofTheorem 9 for details. When α ∈ ( n/ , n ] , Du and Zhang [15] proved thebest known sufficient condition for (2) to hold: s > n n + 1) ( n + 1 − α ) . (3)As mentioned, this is optimal (up to the endpoint) when α = n , but itis not clear yet whether this is optimal for α strictly smaller. It is worthmentioning that (3) is necessary for the α -a.e. pointwise convergence in theperiodic setting [16], however in this setting it is still unknown if it is sufficient(not even for α = n ).In [23] it was proved that for (3 n + 1) / ≤ α ≤ n the condition s > n n + 1) + n − n + 1) ( n − α ) , (4)is necessary for (2) to hold. Here we extend this result to the full range n/ <α ≤ n (recall that for smaller α the problem has been solved in [2]); thus2he result is new for n/ < α ≤ (3 n + 1) / . To prove this result, we use amodification of the Bourgain counterexample rather than the counterexamplein [23]. We consider this fact of independent interest. The possibility ofadapting the Bourgain counterexample to the fractal measure setting wasalso suggested by Lillian Pierce in [27]. Theorem 1.
Let n ≥ and n/ < α ≤ n . Then for every s ′ < s := n n + 1) + n − n + 1) ( n − α ) (5) there exists a function f ∈ H s ′ ( R n ) such that lim sup t → + | e it ~ ∆ / f ( x ) | = ∞ (6) for x in a set of Hausdorff dimension ≥ α . For α ∈ ( n/ , n ) we can in fact immediately improve the statement, sayingthat (6) occurs on a set with α -Hausdorff measure = ∞ . This is because in(5) we have a strict inequality. Thus, given α ′ > α and sufficiently close to α in such a way that s ′ < s := n n + 1) + n − n + 1) ( n − α ′ ) , we would in fact prove that (6) occurs on a set of dimension ≥ α ′ . When α = n we can not self-improve the statement, however we know by [23] that (6)holds on a set of strictly positive Lebesgue measure.A consequence of Theorem 1 is the necessity of the condition s ≥ n n + 1) + n − n + 1) ( n − α ) for the validity of the maximal estimate Z B R sup t ∈ (0 , | e it ∆ f ( x ) | dµ ( x ) . C µ R s k f k , (7)where B R ⊂ R n is a ball of radius R > , and µ is an α -dimensional measureon B R ⊂ R n , i.e. a positive Borel measure that satisfies µ ( B r ( x )) . C µ r α , x and radius r > . One may see (7) as the weighted L inequality Z B R sup t ∈ (0 , | d gdσ ( x ) | dµ ( x ) . C µ R s k g k L ( S ) , (8)where S is a bounded hypersurface in R d := R n +1 with non zero gaussiancurvature (for instance, a portion of the paraboloid in the case of (7)) and dσ is the measure induced on S by the Lebesgue measure. A closely relatedfamily of weighted L estimates is Z B | d gdσ ( Rx ) | dµ ( x ) . C µ R − γ k g k L ( S ) , (9)where B is now a ball in R d of radius , and µ is an α -dimensional measureon B ⊂ R d . The problem here is to identify the largest γ such that (9) holds.Interestingly, these problems are very sensitive to the arithmetical structureof the hypersurface S . For instance, the known necessary conditions aredifferent for the sphere and the paraboloid; see [21, 1, 25, 12, 28, 19]. Notations • e ( z ) = e iz .• If A ⊂ R n , then | A | is its Lebesgue measure, and if A is a discrete set,then | A | is the cardinality. For example, if I = [ a, b ] ⊂ Z denotes theinterval of integers a ≤ k ≤ b , then | I | is the length of the interval.• If I = [ a, b ] ⊂ Z , for a, b ∈ R , denotes an interval of integers, then wewrite L ( I ) := min k ∈ I k and R ( I ) := max k ∈ I k .• B r ( x ) ⊂ R n is a ball of radius r and center x —the center is usuallyomitted. Q ( x, l ) ⊂ R n is a cube with side-length l and center x .• If x . y , then x ≤ Cy for some constant C > , and similarly for x & y ; if x ≃ y then x . y . x . If x ≪ y then x ≤ cy , where c is asufficiently small constant, and similarly for x ≫ y .• lim sup k →∞ F k := T N ≥ S k ≥ N F k .4 Hausdorff dimension of a set: for < α ≤ n and δ > we define theouter measure H αδ ( F ) := inf { X B r ∈B r α | F ⊂ [ B r ∈B B r and r < δ } ; we do not exclude the case δ = ∞ . The α -dimensional Hausdorff mea-sure of a set F is H α ( F ) := lim δ → H αδ ( F ) . The Hausdorff dimensionof a set F is sup { α | H α ( F ) > } . Acknowledgments
This research is funded by the Basque Government through the BERC 2018-2021 program, and by the Spanish State Research Agency through BCAMSevero Ochoa excellence accreditation SEV-2017-0718 and by the IHAIPproject PGC2018-094528-B-I00. Additionally, the first author is supportedby Ikerbasque and the second author is supported by the ERCEA AdvancedGrant 2014 669689-HADE.
We recall some classic estimates about exponential sums that we will userepeatedly in the rest of the paper.We recall first a classical result about Gauss quadratic sums, whose proofcan be consulted in Lemma 3.1 of [27].
Lemma 2 (Gauss quadratic sums) . If a, b, q ∈ Z satisfy the conditions ( a, q ) = 1 and b ∈ Z when q is an odd number, b is even when q ≡ mod ,b is odd when q ≡ mod , (10) then for the quadratic phase f ( r ) := aq r + bq r (11)5 t holds that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q − X r =0 e (2 πf ( r )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = c q √ q, (12) where c q = 1 when q is odd, and c q = √ when q is even. The following estimate due to Weyl will be useful to handle incompleteGauss sums.
Lemma 3.
Let I be an integer interval. If a, b, q ∈ Z satisfy the conditions ( a, q ) = 1 and (10) , then for the quadratic phase f in (11) it holds that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X k ∈ I e (2 πf ( k )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = C | I |√ q + O ( p q ln q ) , (13) where ≤ C ≤ √ .Proof. We can assume that L ( I ) := min k ∈ I k = 0 . In fact, R ( I ) X k = L ( I ) e (2 π ( aq k + bq k )) = e (2 π ( aq L ( I ) + bq L ( I ))) | I |− X k =0 e (2 π ( aq k + b + 2 aL ( I ) q k )) , and the absolute value at both sides is the same; we observe that the parityof b and b + 2 aL ( I ) is preserved.If | I | < q , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X k ∈ I e (2 πf ( k )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C p q ln q ; (14)for the proof we refer to Lemma 3.2 of [27].If | I | ≥ q , then we can sum in blocks of length q . Let M be the largestinteger that satisfies M q ≤ | I | , i.e. M q ≤ | I | < ( M + 1) q , then I = [0 , M q − ∪ J = (cid:16) M − [ m =0 [ mq, mq + q − (cid:17) ∪ J, where | J | < q . The sum over each block [ mq, mq + q − is a Guass quadraticsum, and we arrive to X k ∈ I e (2 πf ( k )) := M q − X r =0 e (2 πf ( r )) + X k ∈ J e (2 πf ( k )) .
6y our election of M we have M = C | I | /q , for < C ≤ , and by (14) wehave (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X k ∈ I e (2 πf ( k )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = C | I | q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q − X r =0 e (2 πf ( r )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + O ( p q ln q ) . Finally, we apply Lemma 2 to get (13).To deal with perturbations of quadratic sums, we will use the followingLemma, which is consequence of Abel’s summation formula; see Lemma 2.3of [16].
Lemma 4.
Let I be an integer interval. Let a k ≥ be a sequence of realnumbers and b k be sequences of complex numbers such that1. a k +1 ≤ a k , (cid:12)(cid:12)P k ∈ I ′ b k (cid:12)(cid:12) ≤ C , for every interval I ′ ⊆ I .Then, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X k ∈ I ′ a k b k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C a L ( I ′ ) , for every interval I ′ ⊆ I. (15) If (1) is replaced with a k +1 ≥ a k , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X k ∈ I ′ a k b k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C a R ( I ′ ) , for every interval I ′ ⊆ I. The initial data we consider are modifications of the Bourgain’s counterex-ample in [5]. Let ϕ be a smooth positive function such that supp ˆ ϕ ⊂ B (0) and ϕ (0) = 1 . We define the function f D ( x ) := f ( x ) ˜ f (˜ x ) (16)where f ( x ) = e (2 πRx ) ϕ ( R x ) , ˜ f (˜ x ) := n Y j =2 ϕ ( x j ) (cid:16) X R D Let c ≪ and let q > be an integer such that RDq ≫ √ ln q .If f is the initial datum (16) , then | e it ~ ∆ / f D ( x ) |k f D k L & R (cid:16) RDq (cid:17) n − (18) for ( x, t ) such that < t = 2 p / ( D q ) ≪ /R and x ∈ E q,D ∩ [0 , c ] n , (19) where E q,D is the set of points x ∈ p RqD + [ − cR − , cR − ] and x j ∈ p j Dq + [ − cR − , cR − ] , ≤ j ≤ n ; (20) here ( p /q, . . . , p n /q ) is an admissible fraction in the sense of Definition 5;see Fig. 1.Proof. If ˆ f is an integrable functions, the solution of the Schrödinger equationwith initial datum f can be represented as e it ~ ∆ / f ( x ) = Z ˆ f ( ξ ) e ( − πt | ξ | + 2 πx · ξ ) dξ. We want to compute the modulus of e it ~ ∆ / f D ( x ) in the region | x | < c and < t < c/R . We note that | e it ~ ∆ / f D ( x ) | = | e it ~ ∆ / f ( x ) | | e it ~ ∆ / ˜ f (˜ x ) | . E q,D in Theorem 6. Some slabs may disappear to satisfy theconditions of admissibility.A direct computation shows that for | t | ≤ c/R and x ∈ tR + [ − cR − , cR − ] (21)we have | e it ~ ∆ / f ( x ) | ≃ | ϕ ( R ( x − tR )) | ≃ (22)Again, a direct computation gives ( ˜ x ∈ R n − ) e it ~ ∆ / ˜ f (˜ x ) = n Y j =2 Z ˆ ϕ ( ξ j ) e ( − πtξ j + 2 πx j ξ j ) X R D On the other hand, the functions φ i ( · ) ( l j ) are real valued, positive, increasingin l j , and satisfy φ i ( · ) ( l j ) . | D || l j || ε j − tξ j | . c, c ≪ recall that | l j | ≤ RD , | ε j | ≤ cR , | t | ≤ cR and | ξ j | ≤ in the support of ˆ φ j .Thus (29) follows by the second part of Lemma 4 taking a l j = φ i ( · ) ( l j ) and b l j = e ( − π p q l j + 2 π p j q l j ) , and the proof is concluded. According to Theorem 6, the function sup If the Fourier transform of ϕ ∈ S ( R ) is supported in ( − , ,then for every N ≥ it holds | e it ~ ∆ / ϕ ( x ) | ≤ C N | x | N , for | x | > t. (39)13 roof. We use the principle of non-stationary phase. We assume that x > t ;the other case is similar. The solution is e it ~ ∆ / ϕ ( x ) = Z ˆ ϕ ( ξ ) e ( − πt | ξ | + 2 πxξ ) dξ. Since ∂ ξ e ( − πt | ξ | + 2 πxξ ) = − πi ( tξ − x ) e ( − πt | ξ | + 2 πxξ ) , then by repeatedintegration by parts we obtain | e it ~ ∆ / ϕ ( x ) | ≤ C N | x − t | N , which is the statement of the Lemma.Before proving the main result of this section, we need to make an ob-servation on the way we define solutions. For f ∈ H s , we define solutionsfor Sobolev functions, in such a way that they are well defined on sets withlarge Hausdorff dimension. Recall that Q ( N ) is the cube of side N centeredat zero. We set e it ~ ∆ / f ( x ) = lim N →∞ S N ( t ) f ( x ) , (40)where S N ( t ) f ( x ) = Z Q ( N ) ˆ f ( ξ ) e ( − πt | ξ | + 2 πx · ξ ) dξ. (41)The limit (40) is usually taken with respect to the L norm, but here we takeall the limits pointwise at each point x where they exist. When f ∈ L ( R ) ,it is known that the limit exists pointwise for almost every x ∈ R and thatit coincides with the L –limit. When n = 1 , this result is due to Carleson[7], whose proof extends to higher dimensions as proved, for instance, in [18].Moreover, we can show that this limit exists γ -almost everywhere for every f ∈ H s with s ∈ (0 , n/ , as long as γ > n − s ; see the appendix of [16].This can be regarded as a refinement of Carleson’s result, although it doesnot recover it. Theorem 9. If g a,b is the initial datum defined in (36) , then lim sup t → + | e it ~ ∆ / g a,b ( x ) | = ∞ (42) for every x ∈ ( F ∩ ([ c , c ] × [0 , c ] n − )) \ Ω , where • c := c , c ≪ c ≪ ; e it ~ ∆ / h k concentrate.• F is a ( a, b ) -set of divergence; • H γ (Ω) = 0 for γ > n − s .Proof. We define h k := kR − sk f D k / k f D k k , where R k := 2 k . From the proof ofTheorem 6 we know that for t = 2 p / ( D k q ) ≪ /R k the value of the solutionat x ∈ F k ∩ [0 , c ] n = S ≤ q ≤ Q k E q,D k ∩ [0 , c ] n is | e it ~ ∆ / h k ( x ) | & k. (43)We fix k ∗ ≥ k ≫ and x ∈ F k ∗ , and we know that x = tR k ∗ + O ( R − k ∗ ) , c R − k ∗ < t < c R − k ∗ . (44)It suffices to prove | e it ~ ∆ / h k ( x ) | . R − k , for k = k ∗ , (45)because then for t = 2 p / ( D k ∗ q ) ≪ /R k ∗ we would have, for all k ≥ k ∗ ,the following (recall (41)) | S k ( t ) g a,b ( x ) | > | e it ~ ∆ / h k ∗ ( x ) | − X k ≤ k = k ∗ ≤ k | e it ~ ∆ / h k ( x ) | & k ∗ , (46)15s long as k ≫ ; then in order to deduce (42) we note that for all x ∈ F ∩ ([ c , c ] × [0 , c ] n − ) we can choose any k ∗ ≥ k ≫ , and we we have alower bound as (46) and the sequence of times t = 2 p / ( D k ∗ q ) ≪ /R k ∗ goesto zero as k ∗ → ∞ . More precisely, since we have e it ~ ∆ / f ( x ) = lim N →∞ S N ( t ) g a,b ( x ) = lim k →∞ S k ( t ) g a,b ( x ) (47)except possibly on sets Ω t , t = 2 p / ( D k ∗ q ) with H γ (Ω t ) = 0 and these setsare countably many, then (42) would follow by (46)-(47), taking Ω := [ t =2 p / ( D k ∗ q ) Ω t . It remains to prove (45). From (23), we see that we can bound e it ~ ∆ / ˜ h k (˜ x ) with the crude estimate | e it ~ ∆ / ˜ f D k (˜ x ) | . (cid:16) R k D k (cid:17) n − , so we can control each term e it ~ ∆ / h k ( x ) as | e it ~ ∆ / h k ( x ) | ≤ | [ e itR k ~ ∆ / ϕ ( R k ( x − tR k ))] | kR k (cid:16) R k D k (cid:17) n − R − sk . | e itR k ~ ∆ / ϕ ( R k ( x − tR k )) | kQ n − k , and we can apply Lemma 8 to ϕ .We verify the hypotheses of Lemma 8 when R k < R k ∗ . By (44) we get R k ( x − tR k ) tR k = R k ( t ( R k ∗ − R k ) + O ( R − k ∗ )) tR k & R − k R k ∗ > , for k, k ∗ ≥ k ≫ ; hence, | e it ~ ∆ / h k ( x ) | . N kQ n − k R − N k . R − k , for N ≫ .We verify now the hypotheses of Lemma 8 when R k > R k ∗ : R k ( tR k − x ) tR k = R k ( t ( R k − R k ∗ ) + O ( R − k ∗ )) tR k & R k > , for k, k ∗ ≥ k ≫ ; hence, | e it ~ ∆ / h k ( x ) | . N kQ n − k ( R k ∗ R − k ) N . R − k , for N ≫ . 16 Dimension of the Divergence Set In the previous section we constructed initial data parameterized by a and b .To simplify matters, we choose those values of a and b for which computationsare easier and exhaust all possible outcomes. Our choices are: ( I ) 12 < a ≤ and b = 2 a − (48) ( II ) 34 < a ≤ and b = 12 . (49)We refer to these ( a, b ) -sets of divergence (Definition 7) as of type I and typeII. We remark that for I we have Q = 1 , and that a = 1 and b = isBourgain’s example. Theorem 10. Let < c ≤ . If F = lim sup k →∞ F k is a ( a, b ) -set ofdivergence (Definition 7), then dim( F ∩ [0 , c ] n ) ≤ α := + ( n − a + b .Proof. Fix a scale < λ ≪ and choose k ′ such that R − k ′ < λ . Since F k is union of . R ( n − a + bk slabs with dimensions R − k × R − k × · · · × R − k , andeach slab can be covered by R k balls B r , for r = R − k , then we can find acollection B k with |B k | = R αk of balls with radius R − k covering F k , so that H βλ ( F ) := inf { X B ρ ∈B ρ β | F ⊂ [ B ρ ⊂B B ρ and ρ < λ } ≤ X k ≥ k ′ X B r ∈B k R − βk , and the last sum is smaller than P k ≥ k ′ R α − βk , which tends to zero as k ′ → ∞ whenever β > α .To prove the corresponding lower bound of dim F , we employ the tech-niques in Section 4 of [24]. We recall a result of Falconer, which is consequenceof Theorem 3.2 and Corollary 4.2 in [17]. Lemma 11. Let < c ≤ . Suppose that there exists a constant C > such that, for all δ > and all cubes Q ( x, δ ) ⊂ [0 , c ] n , we have the densitycondition lim inf k →∞ H β ∞ ( F k ∩ Q ( x, δ )) ≥ Cδ β , where { F k } k ≥ is a sequence of open subsets of B (0 , . Then, for all β ′ < β , H β ′ (lim sup k →∞ F k ) > . 17e prove now the lower bound of dim F in the easier case, in the case ofsets of type I. Theorem 12. If F = lim sup k →∞ F k is a set of type I, that is, < a ≤ and b = 2 a − , then dim F ∩ [0 , c ] n ≥ α where α := 12 + ( n − a + b. (50) Proof. From Lemma 11 it will be sufficient to show that H β ∞ ( F k ∩ Q ( x, δ )) ≥ Cδ β , ∀ Q ( x, δ ) ⊆ [0 , c ] n , (51)holds for all k sufficiently large, where β = α − ε for < ε ≪ . The size of k for which (51) holds will depend on δ . To prove (51) we define an auxiliarymeasure which is a uniform mass measure over F k ∩ Q ( x, δ ) , namely µ k ( A ) := | A ∩ F k ∩ Q ( x, δ ) || F k ∩ Q ( x, δ ) | . Note that µ k depends on the set F k ∩ Q ( x, δ ) , but we will only stress thedependence on k in the notation.Assume we have proved µ k ( B r ) ≤ Cr β δ − β (52)for all sufficiently large k (the size of k will depend on δ ). Using (52) we canprove (51) easily, noting that if B is a collection of balls B r that covers F k ,then µ k ( F k ∩ Q ( x, δ )) ≤ X B r ∈B µ k ( B r ) ≤ Cδ − β X B r ∈B r β . Thus we have reduced to prove (52). To do so we have to work at severalscales. It will be useful to keep in mind that if k ≫ then | F k ∩ Q ( x, δ ) | ≃ R α − nk δ n (53)and that R k → ∞ as k → ∞ . Many estimates below will be indeed justifiedtaking k large enough, depending on δ .18. Scale r < R − k In the worst case a ball is entirely contained in a slabfrom F k , so µ k ( B r ) . r n R − α + nk δ − n ≤ r α δ − n = r β δ − β r α − β δ β − n < r β δ − β R − ( α − β ) k δ β − n ; since α − β > and r < R − k we have R − ( α − β ) k δ β − n < for k ≫ δ thus (52) holds at this scale.2. Scale R − k < r < R − ak . Recall that R − ak < R − k , so a ball B r cannotcontain a slab. On the other hand, since r < R − ak a ball B r intersectsat most one slab, so µ k ( B r ) . rR − ( n − k R − α + nk δ − n = rR − α +1 k δ − n = r α R − ( α − k r α − δ − n < r α δ − n , using R − k < r and α > . Using also r < R − ak we see that µ k ( B r ) . r β R a ( β − α ) k δ − n < r β δ − β , k ≫ δ . 3. Scale R − ak < r < R − k . A ball B r intersects . R ( n − ak r n − slabs, so µ k ( B r ) . r n R ( n − a − n +1 k R − α + nk δ − n ≤ r n R − b + k δ − n . where we used (50). Since r < R − k we have that µ k ( B r ) . r β R β − n − b + k δ − n < r β R ( β − α ) k δ − n where we used α := ( n − a + b + 12 = n − b + n b − < n + 2 b − . (54)Thus µ k ( B r ) < r β δ − β , k ≫ δ . 4. Scale R − k < r < R − bk . A ball B r contains . R ( n − ak r n − slabs, sorecalling again (50) we get µ k ( B r ) . r n − R ( n − a − n + k R − α + nk δ − n = r n − R − bk δ − n < r n − b δ − n , where we used R − k < r . From r < R − bk and (54) we have that µ k ( B r ) . r β R − b ( n +2 b − − β ) k δ − n < r β R − b ( α − β ) k δ − n < r β δ − β , k ≫ δ . ) fractions are very crowded over this lineb) Figure 3: (a) When n = 2 the fractions are already well separated; Lemma 13is unnecessary. (b) When n ≥ the fractions might concentrate aroundsome regions, which prohibits the Frostman measure technique we used inLemma 12. .5. Scale R − bk < r < δ . A ball intersects . R ( n − a + bk r n slabs, so µ k ( B r ) . r n R ( n − a + b − n + k R − α + nk δ − n = r n δ − n < r β δ − β . The inequality (52) thus holds, and so the statement of the Theorem.The lower bound for type II sets is harder to prove, and we need a Lemmathat assures us that for all F k we can find a large sub-collection of slabsuniformly distributed. Similar arguments were used in Lemma 4.3 of [16]and in Sections 5.6–5.8 of [27]. Lemma 13. Let F = lim sup k →∞ F k be a set of type II, that is, < a ≤ and b = . If A k is the collection of slabs in F k ∩ Q ( x, δ ) , for δ < , then,for every ε > and k ≫ ε , we can extract a sub-collection of slabs A ′ k ⊂ A k such that i) |A ′ k | & R − εk |A k | .(ii) If x = ( x , ˜ x ) and y = ( y , ˜ y ) are the centers of two slabs in A ′ k and ˜ x = ˜ y , then | ˜ x − ˜ y | & / ( Q nn − k D k ) .Proof. The sets F k := S s ∈A k s have a periodic structure. In fact, recall thatthe centers of the slabs are (2 p R k / ( qD k ) , p / ( D k q ) , . . . , p n / ( D k q )) , where ( p /q, . . . , p n /q ) is an admissible fraction (Definition 5); hence, F k ismade up of translation of the slabs in the unit cell [0 , R k /D k ] × [0 , /D k ] n − .We assume that k is so large that the number of unit cells not entirelycontained in Q ( x, δ ) is negligible. Therefore, the number of slabs in Q ( x, δ ) is |A k | ≃ D n +1 k R − k δ − n |{ slabs per unit cell }| , and the Lemma reduces to extracta large number of admissible fractions in [0 , n with denominator ≤ Q k . unit cell We drop the subscript k ≫ . Let A be the set of admissible fractions,and let A ⊂ A be the collection of fractions ( p /q, . . . , p n /q ) with q ≡ mod and p j even for ≤ j ≤ n , so that |A | ≃ |A | .We denote by P A the projection of A into the plane ( x , . . . , x n ) , so P A is the set of fractions ( p /q, . . . , p n /q ) with q ≡ mod and even p j . The Dirichlet’s approximation Theorem asserts that for y ∈ R n − there21xists ( p ′ , . . . , p ′ n ) ∈ Z n − such that | y − p ′ j q ′ | ≤ q ′ ( Q/ n − , for some ≤ q ′ ≤ Q/ , (55)so if we write q = 4 q ′ and p j = 2 p ′ j , then we can assert that for every y ∈ R n − there exists a fraction ( p /q, . . . , p n /q ) , for q ≡ mod and p j even, suchthat | y − p j q | ≤ n +1 n − qQ n − , for some ≤ q ≤ Q. In general, a point y ∈ [0 , n − cannot be sufficiently well approximatedby fractions if it satisfies (55) with a fraction ( p ′ /q ′ , . . . , p ′ n /q ′ ) with small q ′ ,so it is convenient to ignore those points. The volume in [0 , n − occupiedby those undesirable points is less than X ≤ q ′ ≤ Q/ n +2 (cid:16) q ′ ( Q/ n − (cid:17) n − (2 q ′ ) n − = 12 . (56)Let G := { y ∈ [0 , n − | y satisfies (55) for some Q/ n +2 < q ′ ≤ Q/ } ,then by (56) the volume of G is > . Cover G with cubes Q ( y, l ) , where y ∈ G and l := 2 n +2+ n − /Q nn − . By Vitali’s covering Theorem we can find adisjoint collection of cubes { Q ( y j , l ) } ≤ j ≤ N such that G ⊂ N [ j =1 Q ( y j , l ); hence, N ≥ c n Q n . We pick from within each Q ( y j , l ) a fraction and constructso a collection of fractions C ⊂ P A ; we define A ⊂ A as the set of fractionssuch that P A = C . By construction, | P A | & Q n and any two points in P A lie at distance & /Q nn − ; the latter, after dilation by /D , implies thecondition (ii) .The fractions in A that lie over ( p /q, . . . , p n /q ) ∈ P A is in numberat least ϕ ( q ) , where ϕ is the Euler’s totient function. Since ϕ ( q ) ≥ q − ε for every ε > and q ≫ ε —see Theorem 327 in [20]—then the number offractions in A is ≥ Q − ε | P A | & Q n +1 − ε ≃ Q − ε |A | , where A is the set ofadmissible fractions; this concludes the verification of condition (i) .22 heorem 14. Let < c ≤ . If F = lim sup k →∞ F k is a set of type II, thatis, < a ≤ and b = , then dim F ∩ [0 , c ] n ≥ α where α := 1 + ( n − a. (57) Proof. We use the same method as in Theorem 12. For fixed ε > , let A ′ k bethe collection of slabs provided by Lemma 13, and let F ′ k be the correspondingset. Given Q ( x, δ ) ⊆ [0 , c ] n , we define again a measure µ k on F k ∩ Q ( x, δ ) that will be useful in the proof; the measure is µ k ( A ) := | A ∩ F ′ k ∩ Q ( x, δ ) || F ′ k ∩ Q ( x, δ ) | . If k ≫ ε then | F ′ k ∩ Q ( x, δ ) | & R α − n − εk δ n . We take β := α − nε < α − ε . The goal is again to prove (52), from whichwe deduce Theorem 14 proceeding as we did in the proof of Theorem 12.Since b = , we can think of the slabs over ( p / ( qD k ) , . . . , p n / ( qD k )) as asingle tube of length 1.1. Scale r < R − . In the worst case a ball is entirely contained in a slabfrom F k , so µ k ( B r ) . r n R − α + n + εk δ − n ≤ r α δ − n = r β δ − β ( r α − β − ε δ β − n ); since α − β > ε and r < R − k , then µ k ( B r ) < r β δ − β whenever k ≫ δ .2. Scale R − k < r < R − ak . By the properties of separation of the slabsin A ′ k , a ball B r intersects at most one slab—recall Lemma 13(ii) and(31)—so µ k ( B r ) . rR − ( n − − α + n + εk δ − n = rR − α +1+ εk δ − n < r α − ε δ − n , where we used α > . Since r < R − ak we see that µ k ( B r ) . r β R a ( β − α + ε ) k δ − n , k ≫ δ . 3. Scale R − ak < r < R k /D k = R n − n +1 (2 a − ) − k . A ball intersects . R ( n − ak r n − “tubes” of length 1 and radius R − k , so (recall (57)) µ k ( B r ) . r n R ( n − a − ( n − − α + n + εk δ − n = r n R εk δ − n = r β δ − β ( r n − β R εk δ β − n ); r < R k /D k ≤ R − n +1 k , we see that µ k ( B r ) . r β δ − β , k ≫ δ . 4. Scale R k /D k < r < δ . A ball B r contains ≃ D n +1 k R − k r n translations ofthe unit cell [0 , R k /D k ] × [0 , /D k ] n − . If V is the volume of F ′ k perunit cell, then | B r ∩ F ′ k | ≃ V D n +1 k R − k r n and | Q ( x, δ ) ∩ F ′ k | ≃ V D n +1 k R − k δ n ; hence µ k ( B r ) . r n δ − n < r β δ − β . The inequality µ k ( B r ) ≤ Cr β δ − β holds for k sufficiently large (dependingon δ ), so the proof is complete. We are now ready to prove our statement combining the results from theprevious section. First we take a, b as in (48)-(49) and recall that we havedefined α := 12 + ( n − a + b. (58)Note that we have a bijection between a ∈ (1 / , (which predicts also thevalue of b by (48)-(49)) and α ∈ ( n/ , n ] , which is the range we are interestedin (the case α = n was handled in [5]).First we claim that given any s ′ < s := n n + 1) + n − n + 1) ( n − α ) (59)we can find a solution u ( x, t ) with initial datum u ∈ H s ′ ( R n ) such that lim sup t → + | u ( x, t ) | = ∞ for x ∈ ( F ∩ ([ c , c ] × [0 , c ] n − )) \ Ω , where F is an ( a, b ) -set of divergence, < c := c ≪ and Ω has dimension ≤ n − s . Indeed, it suffices tochoose u := g a,b defined in (36) so that u ∈ H s ′ ( R n ) for s ′ < s := 14 + n − n + 1) ( n − ( n − a − b ); (60)24ee (38)-(37). Since under (58) the inequality (60) becomes (59), then theclaim follows invoking Theorem 9.Thus, to conclude the proof, we need to show that dim (cid:0) ( F ∩ ([ c , c ] × [0 , c ] n − )) \ Ω (cid:1) ≥ α. (61)First, covering ( F ∩ ([ c , c ] × [0 , c ] n − )) with ≃ ( c /c ) n − cubes of side c ,we see as consequence of Theorems 12 and 14 that dim( F ∩ ([ c , c ] × [0 , c ] n − )) ≥ α. On the other hand, we know that dim Ω ≤ n − s (see Theorem 9). Thus,since for our choice (59) of s we have α > n − s when α > n/ , then (61)follows and the proof is concluded. References [1] J.A. Barceló, J.M. Bennett, A. Carbery, A. Ruiz, and M.C. Vilela. Somespecial solutions of the Schrödinger equation. Indiana University Math-ematics Journal , 56(4):1581–1593, 2007.[2] Juan Antonio Barceló, Jonathan Bennett, Anthony Carbery, andKeith M. Rogers. On the dimension of divergence sets of dispersiveequations. Math. Ann. , 349(3):599–622, 2011.[3] J. Bourgain. A remark on Schrödinger operators. Israel J. Math. , 77(1-2):1–16, 1992.[4] J. Bourgain. On the Schrödinger maximal function in higher dimension. Tr. Mat. Inst. Steklova , 280:53–66, 2013.[5] J. Bourgain. A note on the Schrödinger maximal function. J. Anal.Math. , 130:393–396, 2016.[6] Anthony Carbery. Radial Fourier multipliers and associated maximalfunctions. In Recent progress in Fourier analysis (El Escorial, 1983) ,volume 111 of North-Holland Math. Stud. , pages 49–56. North-Holland,Amsterdam, 1985.[7] Lennart Carleson. On convergence and growth of partial sums of Fourierseries. Acta Math. , 116:135–157, 1966.258] Lennart Carleson. Some analytic problems related to statistical me-chanics. In Euclidean harmonic analysis (Proc. Sem., Univ. Maryland,College Park, Md., 1979) , volume 779 of Lecture Notes in Math. , pages5–45. Springer, Berlin, 1980.[9] Michael G. Cowling. Pointwise behavior of solutions to Schrödingerequations. In Harmonic analysis (Cortona, 1982) , volume 992 of LectureNotes in Math. , pages 83–90. Springer, Berlin, 1983.[10] Björn E. J. Dahlberg and Carlos E. Kenig. A note on the almost every-where behavior of solutions to the Schrödinger equation. In Harmonicanalysis (Minneapolis, Minn., 1981) , volume 908 of Lecture Notes inMath. , pages 205–209. Springer, Berlin-New York, 1982.[11] Ciprian Demeter and Shaoming Guo. Schrödinger maximal func-tion estimates via the pseudoconformal transformation. Preprint.arXiv:1608.07640.[12] Xiumin Du. Upper bounds for Fourier decay rates of fractal measures. Journal of the London Mathematical Society , 102(3):1318–1336, 2020.[13] Xiumin Du, Larry Guth, and Xiaochun Li. A sharp Schrödinger maximalestimate in R . Ann. of Math. (2) , 186(2):607–640, 2017.[14] Xiumin Du, Larry Guth, Xiaochun Li, and Ruixiang Zhang. Pointwiseconvergence of Schrödinger solutions and multilinear refined Strichartzestimates. Forum Math. Sigma , 6:e14, 18, 2018.[15] Xiumin Du and Ruixiang Zhang. Sharp L estimates of the Schrödingermaximal function in higher dimensions. Ann. of Math. (2) , 189(3):837–861, 2019.[16] D. Eceizabarrena and R. Lucà. Convergence over fractals for the periodicSchrödinger equation. Preprint. arXiv:2005.07581.[17] K. J. Falconer. Classes of sets with large intersection. Mathematika ,32:191–205, 1985.[18] Charles Fefferman. On the convergence of multiple Fourier series. Bull.Amer. Math. Soc. , 77:744–745, 1971.2619] L. Guth, A. Iosevich, Y. Ou, and H. Wang. On Falconer’s distance setproblem in the plane. Invent. Math. , 219(3):779–830, 2020.[20] G. H. Hardy and E. M. Wright. An introduction to the theory of numbers.Edited and revised by D. R. Heath-Brown and J. H. Silverman. With aforeword by Andrew Wiles. 6th ed . Oxford: Oxford University Press,6th ed. edition, 2008.[21] Alex Iosevich and Michael Rudnev. Distance measures for well-distributed sets. Discrete & Computational Geometry , 38(1):61–80, July2007.[22] Sanghyuk Lee. On pointwise convergence of the solutions to Schrödingerequations in R . Int. Math. Res. Not. , pages Art. ID 32597, 21, 2006.[23] R. Lucà and K. M. Rogers. A note on pointwise convergence for theSchrödinger equation. Math. Proc. Camb. Philos. Soc. , 166(2):209–218,2019.[24] Renato Lucà and Keith M. Rogers. Coherence on fractals versus point-wise convergence for the Schrödinger equation. Comm. Math. Phys. ,351(1):341–359, 2017.[25] Renato Lucà and Keith M. Rogers. Average decay of the Fouriertransform of measures with applications. J. Eur. Math. Soc. (JEMS) ,21(2):465–506, 2019.[26] A. Moyua, A. Vargas, and L. Vega. Restriction theorems and max-imal operators related to oscillatory integrals in R . Duke Math. J. ,96(3):547–574, 1999.[27] Lillian B. Pierce. On Bourgain’s counterexample for the Schrödingermaximal function. Preprint. arXiv:1912.10574v4.[28] Felipe Ponce-Vanegas. Examples of measures with slow decay of thespherical means of the Fourier transform. Proc. Am. Math. Soc. ,146(6):2617–2621, 2018.[29] Peter Sjögren and Per Sjölin. Convergence properties for the time-dependent Schrödinger equation. Ann. Acad. Sci. Fenn. Ser. A I Math. ,14(1):13–25, 1989. 2730] Per Sjölin. Regularity of solutions to the Schrödinger equation. DukeMath. J. , 55(3):699–715, 1987.[31] T. Tao. A sharp bilinear restrictions estimate for paraboloids. Geom.Funct. Anal. , 13(6):1359–1384, 2003.[32] T. Tao and A. Vargas. A bilinear approach to cone multipliers. I. Re-striction estimates. Geom. Funct. Anal. , 10(1):185–215, 2000.[33] T. Tao and A. Vargas. A bilinear approach to cone multipliers. II.Applications. Geom. Funct. Anal. , 10(1):216–258, 2000.[34] Luis Vega. Schrödinger equations: pointwise convergence to the initialdata. Proc. Amer. Math. Soc. , 102(4):874–878, 1988.[35] Darko Žubrinić. Singular sets of Sobolev functions.