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

Let c(1), . . . , c(s) be nonzero integers satisfying c(1) + + c(s) = 0. We consider the rational quadratic system c(1)x(1)(2)+ +c(s)x(s)(2) = 0 where x(i) are restricted in subset A of Piatetski-Shapiro primes not exceeding x and corresponding to c. We show that for c is an element of ( 1, min{ s/ s-1 , 29/ 28 } ) , if the system has only K -trivial solutions in A, then |A| < x(1/c)(log x)(-1) (log log log log x)((2-s)/(2c)+epsilon) holds for s >= 7.(c) 2023 Elsevier Inc. All rights reserved.

Automated summary

The article introduces a rational quadratic system and provides an upper bound on the number of solutions under specific conditions.