On Improving Roth's Theorem in the Primes

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 $$\alpha \ll \frac{{\left(\log \log \log N\right)}^6}{\log \log N}.$$

14 pages, to appear in Mathematika

Cited by in corpus (5)

• Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions
• Four variants of the Fourier-analytic transference principle
• Solving equations in dense Sidon sets
• Szemerédi's theorem in the primes
• A Density Increment Approach to Roth's Theorem in the Primes