Let T be a discrete abelian group, and E C T. For F C E, we say that F e 9(E), if for all A, finite subsets of I", 0 / A, A + F n F is finite. Having defined the Banach algebra, A(E) = c(E) n B(E), we prove the following: (i) E C T is a Sidon set if and only if every F e 9(E) is a Sidon set; (ii) E e?(r) is a Sidon set if and only if A(E) = A(E). 0. Introduction. Let L denote a discrete abelian...

Let h,k?2 be integers. We say a set A of positive integers is an asymptotic basis order k if every large enough integer can represented as the sum terms from A. called Bh[g] h in at most g different ways. In this paper we prove existence Bh[1] sets which are bases 2h+1 by using probabilistic methods.

We solve an elementary minimax problem and obtain the exact value of the Sidon constant for sets with three elements {n0, n1, n2}: it is sec(π/2n) for n = max |ni − nj |/ gcd(n1 − n0, n2 − n0).

We give asymptotic sharp estimates for the cardinality of a set of residue classes with the property that the representation function is bounded by a prescribed number. We then use this to obtain an analogous result for sets of integers, answering an old question of Simon Sidon. © 2010 Elsevier Inc. All rights reserved. MSC: 11B

Let /, J be discrete spaces and E c / X /. Then either £ is a K-Sidon set (in the sense of [2, §11]), or the restriction algebra A( E) is analytic. The proof is based on probabilistic methods, involving Slépian's lemma.