Primes and polynomials with restricted digits

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 $q$.