On the Uniform Distribution in Residue Classes of Dense Sets of Integers with Distinct Sums

✍ Scribed by Mihail N Kolountzakis

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 95 KB
• Volume: 76
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1998.2351

✦ Synopsis

A set A [1, ..., N] is of the type B 2 if all sums a+b, with a b, a, b # A, are distinct. It is well known that the largest such set is of size asymptotic to N 1Â2 . For a B 2 set A of this size we show that, under mild assumptions on the size of the modulus m and on the difference N 1Â2 &| A | (these quantities should not be too large), the elements of A are uniformly distributed in the residue classes mod m. Quantitative estimates on how uniform the distribution is are also provided. This generalizes recent results of Lindstro m whose approach was combinatorial. Our main tool is an upper bound on the minimum of a cosine sum of k terms, k 1 cos * j x, all of whose positive integer frequencies * j are at most (2&=) k in size.