INFINITE SIDON SETS CONTAINED IN SPARSE RANDOM SETS OF INTEGERS

摘要

A set S of natural numbers is a Sidon set if all the sums s(1) + s(2) with s(1), s(2) epsilon S and s(1) = s(2) are distinct. Let constants alpha 0 and 0 delta 1 be fixed, and let p(m) = min{1, alpha m (1 vertical bar delta)} for all positive integers m. Generate a random set R subset of N by adding m to R with probability p(m), independently for each m. We investigate how dense a Sidon set S contained in R can be. Our results show that the answer is qualitatively very different in at least three ranges of delta. We prove quite accurate results for the range 0 delta = 2/3, but only obtain partial results for the range 2/3 delta = 1.