Infinite Sidon sequences

TL;DR

This paper introduces a novel constructive method using discrete logarithms to generate infinite Sidon sequences with specific growth rates, extending to B_h sequences for all h ≥ 3, improving upon previous non-constructive existence proofs.

Contribution

A new constructive approach based on discrete logarithms for generating infinite Sidon and B_h sequences with precise asymptotic growth rates.

Findings

Constructed an infinite Sidon sequence with A(x)=x^{ { }√2-1+o(1)}.

Extended the method to produce B_h sequences for all h ≥ 3.

Provided explicit growth rate formulas for the sequences.

Abstract

We present a new method to obtain infinite Sidon sequences, based on the discrete logarithm. We construct an infinite Sidon sequence A, with A(x)= x^{\sqrt 2-1+o(1)}. Ruzsa proved the existence of a Sidon sequence with similar counting function but his proof was not constructive. Our method generalizes to B_h sequences: For all h\ge 3, there is a B_h sequence A such that A(x)=x^{\sqrt{(h-1)^2+1}-(h-1)+o(1)}.