On the ubiquity of Sidon sets

Abstract

A Sidon set is a set A of integers such that no integer has two essentially distinct representations as the sum of two elements of A. More generally, for every positive integer g, a B ₂[g]-set is a set A of integers such that no integer has more than g essentially distinct representation-s as the sum of two elements of A. It is proved that almost all small subsets of {1, 2,…, n} are B ₂[g]-sets, in the sense that if B ₂ [g](k, n) denotes the number of B ₂[g]-sets of cardinality k contained in the interval {1,2,…, n}, then

$$ \lim _{n \to \infty } B_2 \left[ g \right]\left( {k,n} \right)/\left( {\frac{n} {k}} \right) = 1 if k = 0 \left( {n^{g/(2g + 2)} } \right). $$

.

Supported in part by grants from the NSA Mathematical Sciences Program and the PSC-CUNY Research Award Program. This paper was written while the author was a visitor at the (alas, now defunct) AT&T Bell Laboratories in Murray Hill, New Jersey, an excellent research institution that split into AT&T Research Labs and Lucent Bell Labs, and provided another instance of a whole being greater than the sum of its parts.