Roth-type Theorem for high-power system in Piatetski-Shapiro primes (II). arXiv:2401.02210
Preprint, arXiv:2401.02210 [math.NT] (2024).
Summary: We consider the nonlinear system \(c_1p_1^d +c_2p_2^d + \dots + c_s p_s^d = 0\) with \(c_1, c_2,\dots, c_s\in\mathbb Z\) being nonzero and satisfying \(c_1 +c_2 + \dots + c_s = 0\). We show that for \(s\ge 2\lfloor \frac{d^2}2\rfloor+1\) and \(c\in\left(1, 1+c(d,s)\right)\), if the system has only \(K\)-trivial solutions in subset \(\mathcal{A}\) of Piatetski-Shapiro primes up to \(x\) and corresponding to \(c\), then \(|\mathcal{A}| \ll \frac{x^{\frac1c}}{\log x} \)\(\left(\log \log \log \log x\right)^{\frac{2-s}{dc}+\varepsilon}\).
MSC:
| 11B30 | Arithmetic combinatorics; higher degree uniformity |
| 11P32 | Goldbach-type theorems; other additive questions involving primes |
| 11L20 | Sums over primes |
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