Recall that a subset X of a group G is 'product-free' if X2 & AND;X = null , i.e. if xy & ISIN;/ X for all x, y & ISIN;X. Let G be a group definable in a distal structure. We prove there are consta...
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Recall that a subset X of a group G is 'product-free' if X2 & AND;X = null , i.e. if xy & ISIN;/ X for all x, y & ISIN;X. Let G be a group definable in a distal structure. We prove there are constants c > 0 and 5 & ISIN;(0, 1) such that every finite subset X & SUBE;G distinct from {1} contains a product-free subset of size at least 5|X|c+1/|X2|c. In particular, every finite k-approximate subgroup of G distinct from {1} contains a product-free subset of density at least 5/kc. The proof is short, and follows quickly from Ruzsa calculus and an iterated application of Chernikov and Starchenko's distal regularity lemma. & COPY;2023 Elsevier B.V. All rights reserved.
We give a self-contained exposition of the recent remarkable result of Kelley and Meka: if ☆A☆⊆☆{☆1☆,☆…☆ ☆,☆N☆}☆ has no nontrivial three-term arithmetic progressions then ☆|☆A☆|☆≤☆ exp☆ ☆(☆−☆c...
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We give a self-contained exposition of the recent remarkable result of Kelley and Meka: if ☆A☆⊆☆{☆1☆,☆…☆ ☆,☆N☆}☆ has no nontrivial three-term arithmetic progressions then ☆|☆A☆|☆≤☆ exp☆ ☆(☆−☆c☆(☆log☆ ☆N☆)☆1☆∕☆1☆2☆)☆N☆, where ☆c☆>☆0☆ is a constant.
Let G be any group and A be a non-empty subset of G. The h-fold product set of A is defined as A(h) := {a(1) . a(2) ... a(h) : a(1), ..., a(h) is an element of A}. Nathanson considered the concept of an asymptotic app...
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Let G be any group and A be a non-empty subset of G. The h-fold product set of A is defined as A(h) := {a(1) . a(2) ... a(h) : a(1), ..., a(h) is an element of A}. Nathanson considered the concept of an asymptotic approximate group. Let r, l is an element of Z(>0). The set A is said to be an (r, l)-approximate group in G if there exists a subset X in G such that vertical bar X vertical bar <= l and A(r) subset of XA. The set A is an asymptotic (r,l)-approximate group if the product set A(h) is an (r,l)-approximate group for all sufficiently large h. Recently, Nathanson showed that every finite subset A of an abelian group is an asymptotic (r, l')-approximate group (with the constant l' explicitly depending on r and A). In this article, our motivations are three-fold: (1) We give an alternate proof of Nathanson's result. (2) From the alternate proof we deduce an improvement in the bound on the explicit constant l'. (3) We generalise the result and show that, in an arbitrary abelian group G, the union of k (unbounded) generalised arithmetic progressions is an asymptotic (r, (4rk)(k))approximate group. (C) 2022 The Authors. Published by Elsevier Inc.
We prove, for a sufficiently small subset A of a prime residue field, an estimate on the number of solutions to the equation (a(1) - a(2))(a(3) - a(4)) = (a(5) - a(6))(a(7) - a(8)) with all variables in A. We then der...
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We prove, for a sufficiently small subset A of a prime residue field, an estimate on the number of solutions to the equation (a(1) - a(2))(a(3) - a(4)) = (a(5) - a(6))(a(7) - a(8)) with all variables in A. We then derive new bounds on trilinear exponential sums and on the total number of residues equaling the product of two differences of elements of A. We also prove a refined estimate on the number of collinear triples in a Cartesian product of multiplicative subgroups and derive stronger bounds for trilinear sums with all variables in multiplicative subgroups.
Let A subset of F-p with vertical bar A vertical bar > 1. We show there is a d is an element of F-p(x) such that d. A contains a gap of size at least 2p/vertical bar A vertical bar-2.
Let A subset of F-p with vertical bar A vertical bar > 1. We show there is a d is an element of F-p(x) such that d. A contains a gap of size at least 2p/vertical bar A vertical bar-2.
For k is an element of N, write S(k) for the largest natural number such that there is a k-colouring of {1, ... , S(k)} with no monochromatic solution to x - y = z(2). That S(k) exists is a result of Bergelson, and a ...
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For k is an element of N, write S(k) for the largest natural number such that there is a k-colouring of {1, ... , S(k)} with no monochromatic solution to x - y = z(2). That S(k) exists is a result of Bergelson, and a simple example shows that S(k) >= 2(2k-1). The purpose of this note is to show that S(k) <= 2(22O(k)).
Improving upon the results of Freiman and Candela-Serra-Spiegel, we show that for a non-empty subset A subset of F-p with p prime and vertical bar A vertical bar 100, then A is contained in an arithmetic progression ...
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Improving upon the results of Freiman and Candela-Serra-Spiegel, we show that for a non-empty subset A subset of F-p with p prime and vertical bar A vertical bar < 0.0045p, (i) if vertical bar A + A vertical bar < 2.59 vertical bar A vertical bar - 3 and vertical bar A vertical bar > 100, then A is contained in an arithmetic progression of size vertical bar A + A vertical bar - vertical bar A vertical bar + 1, and (ii) if vertical bar A - A vertical bar < 2.6 vertical bar A vertical bar-3, then A is contained in an arithmetic progression of size vertical bar A - A vertical bar - vertical bar A vertical bar + 1. The improvement comes from using the properties of higher energies. (C) 2020 Elsevier Inc. All rights reserved.
作者：
Olmezov, K., IState Univ
Moscow Inst Phys & Technol Dolgoprudnyi 141701 Moscow Oblast Russia
We obtain new estimates for the distribution of convolutions of the set of values of a convex function at integer points under additional conditions on the higher derivatives of the function. New estimates for additiv...
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We obtain new estimates for the distribution of convolutions of the set of values of a convex function at integer points under additional conditions on the higher derivatives of the function. New estimates for additive energy and for the dimension of sumsets and sets of differences of such sets arise as natural consequences.
In this paper, we prove some extensions of recent results given by Shkredov and Shparlinski on multiple character sums for some general families of polynomials over prime fields. The energies of polynomials in two and...
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In this paper, we prove some extensions of recent results given by Shkredov and Shparlinski on multiple character sums for some general families of polynomials over prime fields. The energies of polynomials in two and three variables are our main ingredients.
作者：
Mudgal, AkshatPurdue Univ
Dept Math 150 N Univ St W Lafayette IN 47907 USA Univ Bristol
Sch Math Fry BldgWoodland Rd Bristol BS8 1UG Avon England
In this paper, we present a variant of the Balog-Szemeredi-Gowers theorem for the Vinogradov system. We then use our result to deduce a higher degree analogue of the sum-product phenomenon.
In this paper, we present a variant of the Balog-Szemeredi-Gowers theorem for the Vinogradov system. We then use our result to deduce a higher degree analogue of the sum-product phenomenon.
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