Roth's Theorem and the Hardy--Littlewood majorant problem for thin subsets of primes

TL;DR

This paper extends Roth's Theorem and the Hardy--Littlewood majorant property to a broad class of sparse prime subsets, including Piatetski--Shapiro primes, showing they contain infinitely many 3-term arithmetic progressions.

Contribution

It introduces a new class of sparse prime subsets and proves Roth's Theorem and the Hardy--Littlewood majorant property within these sets, generalizing previous results.

Findings

Proves Roth's Theorem for these sparse prime subsets.

Establishes Hardy--Littlewood majorant property for the subsets.

Includes results for Piatetski--Shapiro primes with exponents near 1.

Abstract

We introduce a wide class of deterministic subsets of primes of zero relative density and we prove Roth's Theorem in these sets, namely, we show that any subset of them with positive relative upper density contains infinitely many non-trivial three-term arithmetic progressions. We also prove that the Hardy--Littlewood majorant property holds for these subsets of primes. Notably, our considerations recover the results for the Piatetski--Shapiro primes for exponents close to $1$, which are primes of the form $\lfloor n^c\rfloor$ for a fixed $c>1$.