Roth-type theorem for quadratic system in Piatetski-Shapiro primes

Abstract

Let c1,…,cs be nonzero integers satisfying c1+…+cs=0. We consider the rational quadratic system c1x12+…+csxs2=0 where xi are restricted in subset A of Piatetski-Shapiro primes not exceeding x and corresponding to c. We show that for c∈(1,min⁡{ss−1,2928}), if the system has only K-trivial solutions in A, then |A|≪x1/c(log⁡x)−1(log⁡log⁡log⁡log⁡x)(2−s)/(2c)+ε holds for s⩾7.