The ternary Goldbach problem with two Piatetski-Shapiro primes and a prime with a missing digit

Abstract

Let $${\gamma }^{\ast }=\frac{8}{9}+\frac{2}{3}\frac{\log (10/9)}{\log 10}(\approx 0.919\dots ).$$ Let ${\gamma }^{\ast }<{\gamma }_0\le 1$, $c_0=1/{\gamma }_0$ be fixed. Let also $a_0\in \{0,1,\dots ,9\}$.\\ We prove on assumption of the Generalized Riemann Hypothesis that each sufficiently large odd integer $N_0$ can be represented in the form $$N_0=p_1+p_2+p_3,$$ where the $p_i$ are of the form $p_i=[n_i^{c_0}]$, $n_i\in \mathbb{N}$, for $i=1,2$ and the decimal expansion of $p_3$ does not contain the digit $a_0$.\\ The proof merges methods of J. Maynard from his paper on the infinitude of primes with restricted digits, results of A. Balog and J. Friedlander on Piatetski-Shapiro primes and the Hardy-Littlewood circle method in two variables. This is the first result on the ternary Goldbach problem with primes of mixed type which involves primes with missing digits.