Sparse Admissible Sets and a Problem of Erd\H{o}s and Graham

TL;DR

The paper constructs sparse admissible sets that cannot be translated into primes, answering a question by Erdős and Graham negatively, and introduces multiple new constructions using primitive roots.

Contribution

It provides the first examples of sparse admissible sets that cannot be shifted into primes, using novel greedy and algebraic methods.

Findings

Existence of arbitrarily sparse admissible sets not translatable into primes

Three new constructions of such sets

Answer to Erdős and Graham's question in the negative

Abstract

Erd\H{o}s and Graham asked whether any sparse enough admissible set of natural numbers can be translated into a subset of the primes. By using a greedy construction involving powers of primitive roots, we prove that there exist arbitrarily sparse infinite admissible sets that cannot be translated into a subset of the primes, thus answering this question in the negative. We present three additional constructions as well.