Let A be an ellipsephic set which satis es digital restrictions in
a given base. Using the method developed by Hughes and Wooley, we bound
the number of integer solutions to the system of equations
X2
i=1

x3i
 y3
i
 
=
X5
i=3

x3i
 y3
i
 
X2
i=1
(xi  yi) =
X5
i=3
(xi  yi);
with x; y 2 A5. The fact that ellipsephic sets with small digit sumsets have
fewer solutions of linear equations allows us to improve the general bounds obtained
by Hughes andWooley and also the corresponding e cient congruencing
estimates. We also generalize our result from the curve (x; x3) to (x;  (x)),
where   is a polynomial with integer coe cients and deg( )   3.NSF DMS Grant || NSERC Discovery Grant

Anderson, Theresa C.

Hu, Bingyang

Liu, Yu-Ru

Talmage, Alan

University of Waterloo's Institutional Repository

Bounds on 10th moments of (x, x3) for ellipsephic setsTheresa C. Anderson, Bingyang Hu, Yu-Ru Liu, and Alan TalmageAbstract. Let A be an ellipsephic set which satisfies digital restrictions ina given base. Using the method developed by Hughes and Wooley, we boundthe number of integer solutions to the system of equations2∑i=1(x3i − y3i)=5∑i=3(x3i − y3i)2∑i=1(xi − yi) =5∑i=3(xi − yi),with x,y ∈ A5. The fact that ellipsephic sets with small digit sumsets havefewer solutions of linear equations allows us to improve the general bounds ob-tained by Hughes and Wooley and also the corresponding efficient congruencingestimates. We also generalize our result from the curve (x, x3) to (x, φ(x)),where φ is a polynomial with integer coefficients and deg(φ) ≥ 3.1. introductionThe discrete restriction conjecture has recently been of wide interest for re-searchers in both harmonic analysis and number theory (for example, see [8], [9]2020 Mathematics Subject Classification. 11L07, 42B05, 35Q53.Key words and phrases. mean value estimates, ellipsephic sets, KdV-like equations.T. C. Anderson was supported by an NSF DMS Grant in analysis and number theory.Y.-R. Liu was supported by an NSERC Discovery Grant.The authors would like to thank K. Biggs and T. Wooley for helpful discussions related tothe project. They also thank the referee for providing helpful comments.12 THERESA C. ANDERSON, BINGYANG HU, YU-RU LIU, AND ALAN TALMAGEand [10]). To recall the conjecture for the cuve (x, x3), let a = {a(n)}n∈Z andEa (α, β) :=∑|n|≤Na(n)e(αn3 + βn)with α, β ∈ R.A weaker version of the conjecture can be stated as follows.Conjecture 1.1. For each p ∈ [1,∞] and ε > 0, there exists a constantCp,ε > 0, such that for all N ∈ N and all sequence a = {a(n)}n∈Z ∈ `2(Z), one has‖Ea‖Lp(T2) ≤ Cp,εNε(1 +N12−4p)‖a‖`2(Z).By proving the bound for p = 6, Bourgain [5] established Conjecture 1.1 for thecases 1 ≤ p ≤ 6. Little further progress was made before Hu and Li [8] proved thecase p = 14. Lai and Ding [10] extended the range to p ≥ 12. Recently, Hughes andWooley [9] proved the bound for p = 10 and hence established the conjecture forp ≥ 10. For more details about the discrete restriction conjecture and its relationto KdV equations, we refer interested readers to [8], [9] and [10].Given a set S ⊂ Z, let S(N) = S ∩ [−N,N ] and S = |S(N)|. For s ∈N = {1, 2, · · · }, we denote by Js(S(N)) the number of solutions to the system ofequationss∑i=1(x3i − y3i)= 0s∑i=1(xi − yi) = 0,with x,y ∈ S(N)s. In order to prove Conjecture 1.1 with p = 10, Hughes andWooley studied in [9] the quantity J5(S(N)) for any set S. In particular, theyproved that there exists a positive constant κ such thatJ5(S(N)) N exp(κlogNlog logN)S5.The bound for J5(S(N)) can be improved if additional structures are employed onthe set S. In this paper, we consider the cases of ellipsephic sets.The term ellipsephic set was introduced by Biggs in [3] and [4]. She usedthe terminology to mimic the word ellipséphique used in the French mathematicalBOUNDS ON 10TH MOMENTS OF (x, x3) FOR ELLIPSEPHIC SETS 3literature to denote integers with missing digits (for example, see [1] and [2]). Let` be a positive integer and A` ⊂ {0, 1, · · · , ` − 1} with 2 ≤ |A`| ≤ ` − 1. In otherwords, A` contains at least two elements of {0, 1, · · · , ` − 1}, but at most (` − 1)elements from the set. We say A is an ellipsephic set in the base ` if for any n ∈ A,n =∑iai`i with ai ∈ A`.An ellipsephic set A is said to be with small digit sumsets if there exists someconstantK > 0, such that |A`+A`| ≤ K|A`|. For example, if ` ≥ 3 andA` = {0, 1},then A` +A` ∈ {0, 1, 2} and we have K = 3/2. In this paper, we will show that ifA is an ellipsephic set with small digit sumsets, then we can obtain the followingbound for J5(A(N)).Theorem 1.2. Let A be an ellipsephic set with A` = A ∩ [0, ` − 1]. WriteA(N) = A ∩ [−N,N ] with A = |A(N)|. Suppose that |A` +A`| ≤ K|A`| for someconstant K > 0. Then there exists a positive constant κ such thatJ5(A(N)) K4N4 log`K exp(κlogNlog logN)A6.We recall that for a general set S(N) with S = |S(N)|, Hughes and Wooleyproved in [9, Theorem 2.1] thatJ5(S(N)) N exp(κlogNlog logN)S5.To compare the above result with Theorem 1.2, we first notice that the set S (S`and S, respectively) in [9] plays the role of A (A` and A, respectively) in our setting.Write r for both |S`| and |A`| with 2 ≤ r ≤ `− 1. Then A ≤ rlog`N+1 = rN log` r.It follows that if 4 log`K + log` r < 1, i.e., K4r < `, thenK4N4 log`KA ≤ rK4N4 log`K+log` r = or,K(N).Hence the bound for ellipsephic sets in Theorem 1.2 is sharper in this case. Thecondition K4r < ` is often satisfied. For example, if ` ≥ 3 and A` ∈ {0, 1}, thenr = 2 and K = 3/2. In this case, K4r < ` provided that ` > 2 · (3/2)4, i.e., ` ≥ 11.4 THERESA C. ANDERSON, BINGYANG HU, YU-RU LIU, AND ALAN TALMAGEUsing his efficient congruencing method, Wooley proved in [14] that for a gen-eral set S,J3(S(N)) N εS3for any ε > 0. By trivially bounding the additional four variables by S4, this givesusJ5(S(N)) N εS7.Let r = |S`|. Then S ≤ rN log` r. We notice that the above bound is strongerthan the N1+εS5 bound in [9, Theorem 2.1] provided that S2 = o(N), which holdsif 2 log` r < 1, i.e., r2 < `. By noticing the set S (S` and S, respectively) hereplays the role of A (A` and A, respectively) in Theorem 1.2, we can also comparethe nested efficient congruencing bound N εS7 with the K4N4 log`K+εA6 bound inTheorem 1.2. We see that if 4 log`K < log` r, i.e., K4 < r, thenK4N4 log`K = o(rN log` r),so K4N4 log`K = o(S). Hence the bound in Theorem 1.2 is sharper in this case. Theconditions K4 < r and r2 < ` are often satisfied. For example, if A = {0, 1, · · · , 13},then A+A ∈ {0, 1, · · · , 26}. We have K = 27/14 and r = 14. By taking ` > 142,we have K4 < r < `1/2.Let φ(x) be a polynomial with integer coefficients and deg(φ) ≥ 3. We cangeneralize Theorem 1.2 from the curve (x, x3) to the curve (x, φ(x)). Let J5,φ(A(N))denote the number of solutions to the system of equations5∑i=1(φ(xi)− φ(yi)) = 05∑i=1(xi − yi) = 0,with x,y ∈ A(N)5. We will prove the following result.Theorem 1.3. Let A be an ellipsephic set with A` = A ∩ [0, ` − 1]. WriteA(N) = A∩ [−N,N ] with A = |A(N)|. Suppose that |A` +A`| ≤ K|A`|. Then forBOUNDS ON 10TH MOMENTS OF (x, x3) FOR ELLIPSEPHIC SETS 5any ε > 0, we haveJ5,φ(A(N)) K4N4 log`K+εA6.We assume here and throughout that the implicit constant in the symbol may depend on ε, s, k, and the coefficients of φ.Let |A`| = r with 2 ≤ r ≤ ` − 1. Similar to the remarks after Theorem1.2, the result in Theorem 1.3 is sharper than the general bounds in [2, Theorem3.4] and the corresponding nested efficient congruencing estimates, provided thatK4 < r < `1/2.We now restrict our attention to the case when ` is a prime. A subset R ⊂N ∪ {0} is called a E∗2 -set if for any n ∈ N, we have(1.1) #{(a1, a2) ∈ R2 : a1 + a2 = n} nεfor any ε > 0. Let R` = R ∩ [0, `− 1] and suppose that 2 ≤ |R`| ≤ `− 1. Given aprime ` and a E∗2 -set R, a set E = ER` is called a (`, 2)∗-ellipsephic set ifE ={n =∑iai`i : ai ∈ R` for all i}.In this setting, Biggs [4, Theorem 1.2] proved thatJ5,φ(E(N)) N εE5,where E = |E(N)|. Her bound is essentially optimal as we get J5,φ(E(N))  E5from the diagonal solutions. She also obtained similar bounds for general E∗t -setswith t ≥ 2.The optimal result of Biggs and Theorem 1.3 are applied to sets of differentnature. To illustrate their difference, we first notice that the set E (E` and E,respectively) in [4] plays the role of A (A` and A, respectively) in our setting. In[11], Landau proved that the set of squares satisfies the condition (1.1). Write rfor both |E`| and |A`|. Since the set of squares is sparse, the set E` + E` couldbe of size r2 for sufficiently large `. On the other hand, if an ellipsephic set Asatisfies |A` +A`| < Kr, then Theorem 1.3 provides meaningful improvement only6 THERESA C. ANDERSON, BINGYANG HU, YU-RU LIU, AND ALAN TALMAGEif K4 < r. This condition K < r1/4 is not always satisfied for large r if we take anellipsephic set E with square digits since E` + E` could be of size r2. Thus one cansay the result of Biggs provides useful estimates for “large K,” while Theorem 1.3is meaningful for “small K.”We will prove Theorem 1.2 in Section 2 and Theorem 1.3 in Section 3. The keyidea of our paper is to make use of the fact that ellipsephic sets with small digitsumsets have fewer solutions of linear equations. More precisely, we can boundelements of the form 2A− 2A for ellipsephic sets A with small digit sumsets moreefficiently than general sets (see Lemma 2.1). We will highlight this idea with vari-ations of Theorem 1.3 at the end of the paper.2. Proof of Theorem 1.2To prove Theorem 1.2, we need the following lemma.Lemma 2.1. Let A be an ellipsephic set with A` = A∩ [0, `−1]. Write A(N) =A ∩ [−N,N ] with A = |A(N)|. Suppose that |A` +A`| ≤ K|A`|. Then|2A− 2A|  K4N4 log`KA.Proof: Since A` ⊂ [0, `−1], by viewing A` as a subset of the abelian group Z4`−3 ={2 − 2`, · · · ,−1, 0, 1, · · · , 2` − 2}, we have A` + A` ⊂ Z4`−3. By the Plünnecke-Ruzsa inequality [12], [13], we have |2A` − 2A`| ≤ K4|A`|. Since A ⊂ [−N,N ],each element of A is formed of ≤ log`N + 1 digits, each of which is in A`. Hence,each element of 2A−2A corresponds to at least one element of (2A`−2A`)log`N+1.Thus we have|2A− 2A| ≤ |2A` − 2A`|log`N+1≤ K4(log`N+1)|A`|log`N+1= K4N4 log`KA.BOUNDS ON 10TH MOMENTS OF (x, x3) FOR ELLIPSEPHIC SETS 7Remark Let A be an ellipsephic set satisfying the conditions in Lemma 2.1. Form,n ∈ N, using the same argument as the above proof, we can show that|mA− nA|  Km+nN (m+n) log`KA.LetE1A(N)(α, β) :=∑n∈A(N)e(αn3 + βn).By the orthogonal relation of the exponential function, we see thatJ5(A(N)) =T2∣∣E1A(N)(α, β)∣∣10 dαdβ,where T = [0, 1). Hence to prove Theorem 1, it is equivalent to show that thereexists a positive constant κ such thatT2∣∣E1A(N)(α, β)∣∣10 dαdβ  K4N4 log`K exp(κ logNlog logN)A6.Proof of Theorem 1.2 Write a = 1A(N). The tenth moment ‖Ea‖1010 counts thenumber of solutions to the system of equations2∑i=1(x3i − y3i)=5∑i=3(x3i − y3i)2∑i=1(xi − yi) =5∑i=3(xi − yi),with x,y ∈ A(N)5. By the second equation above, we leth := x1 − y1 + x2 − y2 = x3 − y3 + x4 − y4 + x5 − y5 ∈ 2A− 2A.We now write ‖Ea‖1010 as∑h∈2A−2ATT|Ea(α1, α2)|4 e(−α2h)dα2T|Ea(α1, α3)|6 e(−α3h)dα3dα1.By the triangle inequality and Lemma 2.1, this gives(2.1) ‖Ea‖1010 ≤ K4N4 log`KAT3|Ea(α1, α2)|4 |Ea(α1, α3)|6 dα1dα2dα3.Let ct(k) denote the number of solutions of the simultaneous equationst∑i=1(x3i − y3i ) = k andt∑i=1(xi − yi) = 0.8 THERESA C. ANDERSON, BINGYANG HU, YU-RU LIU, AND ALAN TALMAGEIn the proof of [9, Theorem 2.1], Hughes and Wooley proved thatT3|Ea(α1, α2)|4 |Ea(α1, α3)|6 dα1dα2dα3 ∑|k|≤4N3c2(k)c3(k) exp(κlogNlog logN)A5.It follows that‖Ea‖1010  K4N4 log`KA exp(κlogNlog logN)A5 K4N4 log`K exp(κlogNlog logN)A6.Remark We see in Section 1 a remark after Theorem 1.2 that if K4 < r = |A`|,then the bound of Theorem 1.2 is sharper than the bound derived from the efficientcongruencing method. One can find examples to satisfy Km < r for all m ∈ N,provided that r is sufficiently large. For example, for a large `, if A = {0, 1, · · · , q}with 2q < `, then A+A ∈ {0, 1, · · · , 2q}. We haveK = (2q + 1)/(q + 1) = 2− 1/(q + 1) < 2.Hence by taking q = 2m − 1, we get Km < r = q + 1.3. Proof of Theorem 1.3Let φ(x) be a polynomial with integer coefficients and deg(φ) ≥ 3. LetF1A(N)(α, β) :=∑n∈A(N)e (αφ(n) + βn) .By the orthogonal relation of the exponential function, we see thatJ5,φ(A(N)) =T2∣∣F1A(N)(α, β)∣∣10 dαdβ.Hence to prove Theorem 1.3, it is equivalent to show that for any ε > 0, we haveT2∣∣F1A(N)(α, β)∣∣10 dαdβ  K4N4 log`K+εA6.BOUNDS ON 10TH MOMENTS OF (x, x3) FOR ELLIPSEPHIC SETS 9Proof of Theorem 1.3 Write a = 1A(N). The tenth moment ‖Fa‖1010 counts thenumber of solutions to the system of equations2∑i=1(φ(xi)− φ(yi)) =5∑i=3(φ(xi)− φ(yi))2∑i=1(xi − yi) =5∑i=3(xi − yi),with x,y ∈ A(N)5. Using the same argument as the one in Theorem 1.2, we get(3.1) ‖Fa‖1010 ≤ K4N4 log`KAT3|Fa(α1, α2)|4 |Fa(α1, α3)|6 dα1dα2dα3.Let ct(k) counts the number of solutions of the simultaneous equationst∑i=1(φ(xi)− φ(yi)) = k andt∑i=1(xi − yi) = 0,with x,y ∈ At. In the proof of [9, Theorem 3.4], Hughes and Wooley proved thatthere exists a constant C, depending on φ, such thatT3|Fa(α1, α2)|4 |Fa(α1, α3)|6 dα1dα2dα3 ≤∑|k|≤CNkc2(k)c3(k) N εA5.It follows that‖Fa‖1010  K4N4 log`KAN εA5  K4N4 log`K+εA6.The improved bounds in Theorem 1.2 and Theorem 1.3 come from the factthat we can bound elements of the form 2A− 2A for ellipsephic sets A with smalldigit sumsets more efficiently than general sets. To highlight the idea, we considera variation of Theorem 1.3.Given a set S ⊂ Z, let S(N) = S ∩ [−N,N ] and S = |S(N)|. Suppose that|2S(N)− 2S(N)| ≤ P (S) for some function P of S. Let φ(x) be a polynomial withinteger coefficients and deg(φ) ≥ 3. Following the proof of Theorem 1.3, we have10 THERESA C. ANDERSON, BINGYANG HU, YU-RU LIU, AND ALAN TALMAGETheorem 3.1. Let S(N) ⊂ Z ∩ [−N,N ] with |S(N)| = S. Suppose that|2S(N) − 2S(N)| ≤ P (S) for some function P of S. Then for any ε > 0, wehaveJ5,φ(S(N)) P (S)N εS5.For a set S(N) ⊂ Z∩[−N,N ], suppose that |S(N)+S(N)| ≤ KS for some con-stant K (by Freiman’s theorem [6] [7], sets satisfying this condition are containedin a generalized arithmetic progression). By the Plünnecke-Ruzsa inequality [12],[13], we have |2S(N)− 2S(N)| ≤ K4S. Hence as a direct consequence of Theorem3.1, we haveCorollary 3.2. Let S(N) ⊂ Z ∩ [−N,N ] with |S(N)| = S. Suppose that|S(N) + S(N)| ≤ KS for some constant K. Then for any ε > 0, we haveJ5,φ(S(N)) K4N εS6.References[1] K. Aloui, Sur les entiers ellipséphiques: somme des chiffres et répartition dans les classes decongruences. Period. Math. Hungar. 70 (2015), no. 2, 171–208.[2] K. Aloui, C. Mauduit, and M. Mkaouar, Somme des chiffres et répartition dans les classesde congruences. pour les palindromes ellipséphiques. Acta Math. Hungar. 151 (2017), no. 2,409–445.[3] K. D. Biggs, Efficient congruencing in ellipsephic sets: the quadratic case. Acta Arith. 200(2021), no. 4, 331–348.[4] K. D. Biggs, Efficient congruencing in ellipsephic sets: the general case, 2019. arXiv:1912.04351.[5] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applica-tions to nonlinear evolution equations. Part II: The KDV-equation. Geometric and FunctionalAnalysis 3, no. 3 (1993): 209–62.[6] G. A. Fridman, Addition of finite sets, Soviet Mathematics. Doklady 5: 1366–1370.[7] G. A. Freiman, Foundations of a Structural Theory of Set Addition (in Russian), Kazan:Kazan Gos. Ped. Inst.: p. 140.[8] Y. Hu, and X. Li. Discrete Fourier restriction associated with KdV equations. Analysis andPDE 6, no. 4 (2013): 859–92.BOUNDS ON 10TH MOMENTS OF (x, x3) FOR ELLIPSEPHIC SETS 11[9] K. Hughes and T. D. Wooley, Discrete restriction for (x, x3) and related topics, Int. Math.Res. Not. IMRN, (2021): 20 pages[10] X. Lai and Y. Ding. A note on the discrete Fourier restriction problem. Proceedings of theAmerican Mathematical Society 146, no. 9 (2018): 3839–46.[11] E. Landau. Über die Anzahl der Gitterpunkte in geweissen Bereichen. Nachr. Ges. Wiss.Goettingen, Math.-Phys. Kl. 1912 (1912), 687-770.[12] H. Plünnecke, Eine zahlentheoretische Anwendung der Graphtheorie, Journal für die reineund angewandte Mathematik. 243 (1970): 171–183.[13] I. Ruzsa, An application of graph theory to additive number theory, Scientia, Series A 3(1989): 97–109.[14] T. D. Wooley, Nested efficient congruencing and relatives of Vinogradov’s mean value theo-rem, Proc. London Math. Soc. 118 (2019), no. 4, 942–1016.Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh,PA 15213.E-mail address: tanders2@andrew.cmu.eduDepartment of Mathematics, Purdue University, 150 N. University St., West Lafayette,IN, 47907, U.S.AE-mail address: hu776@purdue.eduDepartment of Pure Mathematics, University of Waterloo, 200 University Ave.West, Waterloo, ON, N2L 3G1, CANADAE-mail address: yrliu@uwaterloo.caDepartment of Pure Mathematics, University of Waterloo, 200 University Ave.West, Waterloo, ON, N2L 3G1, CANADAE-mail address: alan.talmage@uwaterloo.ca

Bounds on 10th moments of (x, x^3) for ellipsephic sets

https://uwspace.uwaterloo.ca/bitstream/10012/20011/1/2023-10th-moment.pdf

Bounds on 10th moments of (x, x^3) for ellipsephic sets

Abstract

Similar works

Full text

Available Versions

University of Waterloo's Institutional Repository