Roth-type Theorem for high-power system in Piatetski-Shapiro primes (II)

2024/01/04 by Xiumin Ren, Ren, Xiumin, Yu‐Chen Sun +5
Mathematics · #Finite Group Theory Research #Analytic Number Theory Research #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.2401.02210

Abstract

We consider the nonlinear system c₁p₁^d +c₂p₂^d + … + c_s p_s^d = 0 with c₁, c₂,…, c_s∈\mathbb Z being nonzero and satisfying c₁ +c₂ + … + c_s = 0. We show that for s≥ 2\lfloor \fracd²2\rfloor+1 and c∈(1, 1+c(d,s)), if the system has only K-trivial solutions in subset A of Piatetski-Shapiro primes up to x and corresponding to c, then |A| ≪ \fracx^\frac1clog x (log log log log x)^{(2-s)/(dc)+ε}.

Related

• Roth's Theorem in the Piatetski-Shapiro primes
• A hybrid of two theorems of Piatetski-Shapiro
• Roth's Theorem and the Hardy--Littlewood majorant problem for thin subsets of primes
• Vinogradov's three primes theorem in the intersection of multiple Piatetski-Shapiro sets
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