Double Character Sums with Intervals and Arbitrary Sets

Shkredov, Ilya D.; Shparlinski, Igor E.

doi:10.1134/s0081543818080199

Cited by 4 publications

(1 citation statement)

References 32 publications

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“…In the case that

A

B

is highly structured, Vinogradov's estimate can be improved. Theorems of this sort can be found in [2–4, 6, 14, 15, 18]. For completely general sets, [1] provides a slight improvement for certain primes, while one also has a very modest improvement if we replace a twofold convolution with a threefold convolution, which means estimating

S_{χ} false(A, B, C false) = \sum_{a \in A} \sum_{b \in B} \sum_{c \in C} χ false(a + b + c false) .

This was carried out by the first author in [7].…”

Section: Introductionmentioning

confidence: 99%

Refined estimates concerning sumsets contained in the roots of unity

Hanson

Petridis

2020

Proc. London Math. Soc.

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We prove that the clique number of the Paley graph is at most a p{2`1, and that any supposed additive decompositions of the set of quadratic residues can only come from co-Sidon sets.Theorem 1.1. If d ą 4 then Z d " A`B has no non-trivial solutions over C.Proof. Assume 2 ď |A| ď |B|. Let a 1 , a 2 P A be distinct. Then, since Z d Ă S 1 (the unit circle in the complex plane) we have that a 1`B and a 2`B are subsets of S 1 . But a 1´a2 " pa 1`b q´pa 2`b q

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“…In the case that

A

B

S_{χ} false(A, B, C false) = \sum_{a \in A} \sum_{b \in B} \sum_{c \in C} χ false(a + b + c false) .

This was carried out by the first author in [7].…”

Section: Introductionmentioning

confidence: 99%

Refined estimates concerning sumsets contained in the roots of unity

Hanson

Petridis

2020

Proc. London Math. Soc.

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Congruences with intervals and arbitrary sets

Banks

Shparlinski

2019

Arch. Math.

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A quadratic Vinogradov mean value theorem in finite fields

Mansfield,

Mudgal

2024

The Quarterly Journal of Mathematics

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Let p be a prime, let $s \geq 3$ be a natural number and let $A \subseteq \mathbb{F}_p$ be a non-empty set satisfying $|A| \ll p^{1/2}$. Denoting $J_s(A)$ to be the number of solutions to the system of equations $$ \sum_{i=1}^{s} (x_i - x_{i+s}) = \sum_{i=1}^{s} (x_i^2 - x_{i+s}^2) = 0, $$ with $x_1, \dots, x_{2s} \in A$, our main result implies that $$ J_s(A) \ll |A|^{2s - 2 - 1/9}. $$ This can be seen as a finite field analogue of the quadratic Vinogradov mean value theorem. Our techniques involve a variety of combinatorial geometric estimates, including studying incidences between Cartesian products A × A and a special family of modular hyperbolae.

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Double Character Sums with Intervals and Arbitrary Sets

Abstract: We obtain a new bound on certain double sums of multiplicative characters improving the range of several previous results. This improvement comes from new bounds on the number of collinear triples in finite fields, which is a classical object of study of additive combinatorics.

Cited by 4 publications

References 32 publications

Refined estimates concerning sumsets contained in the roots of unity

Refined estimates concerning sumsets contained in the roots of unity

Congruences with intervals and arbitrary sets

A quadratic Vinogradov mean value theorem in finite fields

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