Vsevolod F. Lev

Sum-free sets in abelian groups

AbstractWe show that there is an absolute constant δ>0 such that the number of sum-free subsets of any finite abelian groupG is

RESTRICTED SET ADDITION IN GROUPS I: THE CLASSICAL SETTING

\left( {2^{\nu (G)} - 1} \right)2^{\left| G \right|/2} + O\left( {2^{(1/2 - \delta )\left| G \right|} } \right)

Large sum-free sets in ℤ/ p ℤ

whereν(G) is the number of even order components in the canonical decomposition ofG into a direct sum of its cyclic subgroups, and the implicit constant in theO-sign is absolute.

Progression-free sets in finite abelian groups

Abstract. We consider two general principles which allow us to reduce certain additive problems for residue classes modulo a prime to the corresponding problems for integers. <lsiheader> <onlinepub>26 June, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>19n3p343.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>yes <sectionname> </lsiheader>

Sums and differences along Hamiltonian cycles

We survey the existing and prove several new results for the cardinality of the restricted doubling 2 A = {a′+a′′ : a, a ∈ A, a 6= a′′}, where A ⊆ G is a subset of the set of elements of an (additively written) group G. In particular, we improve known estimates for G = Z and G = Z/pZ and give a first-of-a-kind general estimate valid for arbitrary G. 1. Background, motivation and summary of results Let G be an arbitrary group. We use additive notation for the group operation in G, as all particular groups that appear in this paper (excluding the Appendix) are Abelian; however, no commutativity is assumed in general, unless indicated explicitly. Let A ⊆ G and B ⊆ G be finite non-empty subsets of the set of all elements of G. How small can be the set A+B = {a+ b : a ∈ A, b ∈ B} of all elements representable as a sum of an element of A and an element of B? Though this and related problems are studied in numerous papers, almost nothing is known about the cardinality of the set A +B = {a+ b : a ∈ A, b ∈ B, a 6= b} of all sums with distinct summands. We are primarily interested in B = A and we abbreviate 2A = A + A and 2 A = A +A. The first case one might think of is G = Z, the group of integers. Here we can shift A to make its minimum element 0 and then divide through all the shifted elements by their greatest common divisor — this normalization, clearly, does not affect the cardinalities of 2A and 2 A. We denote then by l the maximum element of, and by n the cardinality of A. Thus, there is no loss of generality in writing A ⊆ [0, l], 0, l ∈ A, gcd(A) = 1, |A| = n. It was proved by G. Freiman over 30 years ago (see [5]) that under this notation |2A| ≥ min{l, 2n− 3}+ n = { l + n, if l ≤ 2n− 3, 3n− 3 if l ≥ 2n− 2, 1991 Mathematics Subject Classification. Primary: 11B75; Secondary: 05D99, 20F99. 1

Generating abelian groups by addition only

A well-known result by Kemperman describes the structure of those pairs (A, B) of finite subsets of an abelian group satisfying |A + B| ≤ |A| + |B| -1. We establish a description which is, in a sense, dual to Kempermans, and as an application sharpen several results due to Deshouillers, Hamidoune, Hennecart, and Plagne.

Greedy algorithm, arithmetic progressions, subset sums and divisibility

We show that ifp is prime andA is a sum-free subset of ℤ/pℤ withn:=|A|>0.33p, thenA is contained in a dilation of the interval [n,p−n] (modp).

How long does it take to generate a group

Abstract Let G be a finite abelian group. Write 2G≔{2g : g∈G} and denote by rk(2G) the rank of the group 2G. Extending a result of Meshulam, we prove the following. Suppose that A⊆G is free of “true” arithmetic progressions; that is, a1+a3=2a2 with a1,a2,a3∈A implies that a1=a3. Then |A| As a corollary, we generalize a result of Alon and show that if an integer k⩾2 and a real e>0 are fixed, |2G| is large enough, and a subset A⊆G satisfies |A|⩾(1/k+e)|G|, then there exists A0⊆A such that 1⩽|A0|⩽k and the elements of A0 add up to zero. When G is of odd order or cyclic this reduces to the original result of Alon.