Sidon Basis

TL;DR

This paper advances the understanding of Sidon sequences by proving Erd"os's conjecture for large cyclic groups, constructing infinite Sidon sequences that are asymptotic bases, and exploring sequences with flexible basis orders.

Contribution

It proves Erd"os's conjecture for large cyclic groups and constructs new Sidon sequences with asymptotic basis properties and adjustable order.

Findings

Erd"os conjecture holds for all large cyclic groups Z_N.

Existence of infinite B_2[2] sequences that are asymptotic bases of order 3.

Sidon sequences can be asymptotic bases of order 3+c for any c>0.

Abstract

Erd\"os conjectured the existence of an infinite Sidon sequence of positive integers which is also an asymptotic basis of order 3. We make progress towards this conjecture in several directions. First we prove the conjecture for all cyclic groups Z_N with N large enough. In second place we prove by probabilistic methods that there is an infinite B_2[2] sequence which is an asymptotic basis of order 3. Finally we prove that for all c>0 there is a Sidon sequence which is an asymptotic basis of order 3+c,that is to say, any positive sufficiently large integer n can be written as a sum of 4 elements of the sequence, one of them smaller than n^c.