Primes and polynomials with restricted digits

TL;DR

This paper proves the existence of infinitely many primes and polynomial values avoiding a specific digit in a given base, using advanced Fourier analysis and the Hardy-Littlewood circle method.

Contribution

It establishes new results on primes and polynomial values with digit restrictions in large bases, extending previous work with novel analytical techniques.

Findings

Infinitely many primes exclude a specific digit in base q.

Polynomial values satisfying local conditions can avoid certain digits.

Method combines circle method with Fourier analysis of digit-restricted sets.

Abstract

Let $q$ be a sufficiently large integer, and $a_0\in\{0,\dots,q-1\}$. We show there are infinitely many prime numbers which do not have the digit $a_0$ in their base $q$ expansion. Similar results are obtained for values of a polynomial (satisfying the necessary local conditions) and if multiple digits are excluded. Our proof is based on the Hardy-Littlewood circle method and Fourier analysis of the set of integers with no digit equal to $a_0$ in base $q$.