On Improving Roth's Theorem in the Primes

TL;DR

This paper improves bounds on the density of prime subsets avoiding non-trivial arithmetic progressions, advancing Roth's theorem in the primes by refining previous estimates through modifications of Helfgott and De Roton's work.

Contribution

It introduces a refined bound on the density of prime sets lacking arithmetic progressions, enhancing prior results by modifying existing methods.

Findings

Improved upper bound on the density of prime sets without 3-term arithmetic progressions.

Demonstrated the bound is tighter than previous estimates involving iterated logarithms.

Extended the methodology of Helfgott and De Roton to achieve these improvements.

Abstract

Let $A\subset\left\{ 1,\dots,N\right\} $ be a set of prime numbers containing no non-trivial arithmetic progressions. Suppose that $A$ has relative density $\alpha=|A|/\pi(N)$, where $\pi(N)$ denotes the number of primes in the set $\left\{ 1,\dots,N\right\} $. By modifying Helfgott and De Roton's work, we improve their bound and show that $$\alpha\ll\frac{\left(\log\log\log N\right)^{6}}{\log\log N}.$$