On partitioning Sidon sets with quasi-independent sets

Abstract

There is a construction of random subsets of Z in which almost every subset is Sidon (this was first done by Katznelson). More is true: almost every subset is the finite union of quasi-independent sets. Also, if every Sidon subset of Z\{0} is the finite union of quasi-independent sets, then the required number of quasi-independent sets is bounded by a function of the Sidon constant. Analogs of this last result are proved for all Abelian groups, and for other special Sidon sets (the N -independent sets). Sidon subsets have been characterized by Pisier as having proportional quasiindependent subsets[8]. There remains the open problem of whether Sidon subsets of Z must be finite unions of quasi-independent sets. Grow and Whicher produced an interesting example of a Sidon set whose Pisier proportionality was 1/2 but the set was not the union of two quasi-independent sets [3]. On the other hand, this paper provides probabilistic evidence in favor of an affirmative answer with a construction of random Sidon sets which borrows heavily from ideas of Professors Katznelson and Malliavin [4,5,6]. Katznelson provided a random construction of integer Sidon sets which, almost surely, were not dense in the Bohr compactifaction of the integers [5,6]. This paper presents a modification of that construction and emphasizes a stronger conclusion which is implicit in the earlier construction: almost surely, the random sets are finite unions of quasi-independent sets (also of N -independent sets, defined below). In this paper, random subsets of size O(log nj) are chosen from disjoint arithmetic progressions of length nj (the maximum density allowed for a Sidon set), with nj → ∞ fast enough and the progressions rapidly dilated as j →∞. This paper concludes with several deterministic results. If every Sidon subset of Z\{0} is a finite union of quasi-independent sets, then the required number of quasi-independent sets is bounded by a function of the Sidon constant. Analogs of this result are proved for all Abelian groups, and for other special Sidon sets (the N -independent sets). Throughout this paper, unspecified variables denote positive integers. 1991 Mathematics Subject Classification. 43A56.