Joint ergodicity of fractional powers of primes

Abstract We establish the mean convergence for multiple ergodic averages with iterates given by distinct fractional powers of primes and related recurrence results. A consequence our main result is that every set integers positive upper density contains patterns form $\{m,m+[p_n^a], m+[p_n^b]\}$ , where $a,b$ are nonintegers $p_n$ denotes n th prime, a property fails if or b natural number. Our approach based on recent criterion joint ergodicity collections sequences, bulk proof devoted to obtaining good seminorm estimates averages. The input needed from number theory bounds prime k -tuples follow elementary sieve equidistribution results in circle.