On the Growth of the Counting Function of Stanley Sequences

TL;DR

This paper proves that the counting function of Stanley sequences, generated from sets with no 3-term arithmetic progressions, grows at least proportionally to the square root of x, confirming a conjecture about their growth rate.

Contribution

It establishes a lower bound on the growth of the counting function of Stanley sequences, confirming a conjecture and strengthening previous results.

Findings

S(A,x)  (\u221a{2}-) x for large x

Affirms that Stanley sequences grow at least as fast as a constant times   x

Provides a quantitative growth rate bound for Stanley sequences

Abstract

Given a finite set of nonnegative integers A with no 3-term arithmetic progressions, the Stanley sequence generated by A, denoted S(A), is the infinite set created by beginning with A and then greedily including strictly larger integers which do not introduce a 3-term arithmetic progressions in S(A). Erdos et al. asked whether the counting function, S(A,x), of a Stanley sequence S(A) satisfies S(A,x)>x^{1/2-\epsilon} for every \epsilon>0 and x>x_0(\epsilon,A). In this paper we answer this question in the affirmative; in fact, we prove the slightly stronger result that S(A,x)\geq (\sqrt{2}-\epsilon)\sqrt{x} for x\geq x_0(\epsilon,A).