Roth-type Theorem for High-power System in Piatetski-Shapiro primes

Abstract

Let c₁, …, c_s be non-zero integers satisfying c₁ + ⋯ + c_s = 0. We consider the system $c_{1}x_{1}^{d}+\cdots+c_{s}x_{s}^{d}=0$ with d ≥ 3, where x_i are restricted in subset $\cal{A}$ of Piatetski-Shapiro primes not exceeding x and corresponding to c. We show that for s > S(d)+2 and $c\in(1,1+\tilde{c}(d,s))$, if the system has only K-trivial solutions in $\cal{A}$, then $\vert\cal{A}\vert\ll{x^{1/c}\over{\log x}}(\log\log\log\log x)^{{2-s\over{dc}}+\varepsilon}$, where S(3) = 8, S(4) = 16, S(d) = d(d + 1) (d ≥ 5), and $\tilde{c}(d,s)=\min\{{2d\over{(S(d)+1)(3d-4)(3d-2)(3d+2)-d)-d}},{{d\over{S(d)s-d}}}\}$.

Keywords

Piatetski-Shapiro prime / Roth theorem / transference principle

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