A family A of k-subsets of {1, 2, . . . , N} is a Sidon system if the sumsets A + B, A, B is an element of A are pairwise distinct. We show that the largest cardinality F-k(N) of a Sidon system of k-subsets of [N] sat...
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A family A of k-subsets of {1, 2, . . . , N} is a Sidon system if the sumsets A + B, A, B is an element of A are pairwise distinct. We show that the largest cardinality F-k(N) of a Sidon system of k-subsets of [N] satisfies F-k(N) <= ((N-1)(k-1)) + N - k and the asymptotic lower bound F-k(N) = Omega(k)(Nk-1). More precise bounds on F-k(N) are obtained for k <= 3. We also obtain the threshold probability for a random system to be Sidon for k >= 2.
A finite setA= {a(1)< horizontal ellipsis <a(n)}subset of Double-struck capital R is said to beconvexif the sequence (a(i)-a(i-1))i=2n is strictly increasing. Using an estimate of the additive energy of convex s...
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A finite setA= {a(1)< horizontal ellipsis additive energy of convex sets, one can estimate the size of the sumset as divide A+A divide greater than or similar to divide A divide (102/65), which slightly sharpens Shkredov's latest result divide A+A divide greater than or similar to divide A divide (58/37).
In this paper we consider estimating the number of solutions to multiplicative equations in finite fields when the variables run through certain sets with high additive structure. In particular, we consider estimating...
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In this paper we consider estimating the number of solutions to multiplicative equations in finite fields when the variables run through certain sets with high additive structure. In particular, we consider estimating the multiplicative energy of generalized arithmetic progressions in prime fields and of boxes in arbitrary finite fields. We obtain sharp bounds in more general scenarios than previously known. Our arguments extend some ideas of Konyagin and Bourgain and Chang into new settings. (c) 2021 Published by Elsevier Inc.
A non-quantitative version of the Freiman-Ruzsa theorem is obtained for finite stable sets with small tripling in arbitrary groups, as well as for (finite) weakly normal subsets in abelian groups.
A non-quantitative version of the Freiman-Ruzsa theorem is obtained for finite stable sets with small tripling in arbitrary groups, as well as for (finite) weakly normal subsets in abelian groups.
A set of integers S subset of N is an alpha-strong Sidon set if the pairwise sums of its elements are far apart by a certain measure depending on alpha, more specifically if vertical bar(x + w) - (y + z)vertical bar &...
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A set of integers S subset of N is an alpha-strong Sidon set if the pairwise sums of its elements are far apart by a certain measure depending on alpha, more specifically if vertical bar(x + w) - (y + z)vertical bar >= max{x(alpha), y(alpha), z(alpha), w(alpha)} for every x, y, z, w is an element of S satisfying max{x, w} not equal max{y, z}. We obtain a new lower bound for the growth of alpha-strong infinite Sidon sets when 0 <= alpha < 1. We also further extend that notion in a natural way by obtaining the first non-trivial bound for alpha-strong infinite B-h sets. In both cases, we study the implications of these bounds for the density of, respectively, the largest Sidon or B-h set contained in a random infinite subset of N. Our theorems improve on previous results by Kohayakawa, Lee, Moreira and Rodl. (C) 2021 Elsevier Inc. All rights reserved.
We study the maximum possible size of a subset in a vector space over a finite field which contains no solution of a given linear equation (or a system of linear equations). This is a finite field version of Ruzsa'...
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We study the maximum possible size of a subset in a vector space over a finite field which contains no solution of a given linear equation (or a system of linear equations). This is a finite field version of Ruzsa's work [7]. (C) 2021 Elsevier B.V. All rights reserved.
Let Abe a set of integers which is dense in a finite interval. We establish upper and lower bounds for the longest regularly-spaced and convex sequences in Aand in A - A. (C) 2020 Elsevier Inc. All rights reserved.
Let Abe a set of integers which is dense in a finite interval. We establish upper and lower bounds for the longest regularly-spaced and convex sequences in Aand in A - A. (C) 2020 Elsevier Inc. All rights reserved.
A rank-n binary matroid is a spanning subset E of F n 2 \ {0}, a triangle is a set of three elements from E which sum to zero, and the density of a rank-n binary matroid is |E|/2 n. We begin by giving a new exposition...
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A rank-n binary matroid is a spanning subset E of F n 2 \ {0}, a triangle is a set of three elements from E which sum to zero, and the density of a rank-n binary matroid is |E|/2 n. We begin by giving a new exposition of a result due to Davydov and Tombak, which states that if E is a rank-n triangle-free matroid of density greater than 1/4, then there is a dimension-(n − 2) subspace of F n 2 which is disjoint from E. With this as a starting point, we provide a recursive structural decomposition for all maximal triangle- free binary matroids of density greater than 1/4. A key component of this decomposition is an analogous characterization of matroids which are maximal with respect to containing exactly one triangle. A pentagon in a binary matroid E is a set of 5 elements which sum to zero. We conjecture that if E is a rank-n triangle-free binary matroid, then E contains at most 2 4n−16 pentagons, and provide a potential extremal example. We first resolve this conjecture when E has density at most 4 √ 120/16 ≈ 0.20568. Thereafter, we use our structural decomposition to show that the conjecture holds for matroids with density greater than 1/4. This leaves the interval? 4 √ 120/16, 1/4?, where the conjecture remains unresolved.
We show that there exists an absolute constant c > 0, such that, for any finite set A of quaternions, max{vertical bar A + A vertical bar;vertical bar AA vertical bar} greater than or similar to vertical bar A vert...
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We show that there exists an absolute constant c > 0, such that, for any finite set A of quaternions, max{vertical bar A + A vertical bar;vertical bar AA vertical bar} greater than or similar to vertical bar A vertical bar(4/3+c): This generalizes a sum-product bound for real numbers proved by Konyagin and Shkredov.
A set of integers is sum-free if it contains no solution to the equation x + y = z. We study sum-free subsets of the set of integers [n] = {1,...n} for which the integer 2n + 1 cannot be represented as a sum of their ...
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A set of integers is sum-free if it contains no solution to the equation x + y = z. We study sum-free subsets of the set of integers [n] = {1,...n} for which the integer 2n + 1 cannot be represented as a sum of their elements. We prove a bound of O(2(n/3)) on the number of these sets, which matches, up to a multiplicative constant, the lower bound obtained by considering all subsets of B-n = {inverted right perperndicular2/3 (n + 1)inverted left perpendicular,..., n}. A main ingredient in the proof is a stability theorem saying that if a subset of [n] of size close to vertical bar B-n vertical bar contains only a few subsets that contradict the sum-freeness or the forbidden sum, then it is almost contained in B-n. Our results are motivated by the question of counting symmetric complete sum-free subsets of cyclic groups of prime order. The proofs involve Freiman's 3k - 4 theorem, Green's arithmetic removal lemma, and structural results on independent sets in hypergraphs.
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