Roth-type theorem for nonlinear equations in Piatetski-Shapiro primes

: Ren, Xiumin; Sun, Yu-Chen; Zhang, Qingqing; Zhang, Rui

Publisher: WORLD SCIENTIFIC PUBL CO PTE LTD

: SINGAPORE

: 2024

: International Journal of Number Theory

: INTERNATIONAL JOURNAL OF NUMBER THEORY

: INT J NUMBER THEORY

: 16

: 1793-0421

: 1793-7310

DOI: https://doi.org/10.1142/S1793042125500447

: https://doi.org/10.1142/S1793042125500447

We consider the nonlinear diophantine equation c(1)p(1)(d) + c(2)p(2)(d) + center dot center dot center dot + c(s)p(s)(d) = 0 with c(1),c(2),...,c(s) is an element of Z being non-zero and satisfying c(1) + c(2) + center dot center dot center dot + c(s) = 0. We show that for s >= 2(sic)d(2)/2(sic) + 1 and c is an element of (1, 1 + c(0)(d,s)), if the equation has only K-trivial solutions in a subset epsilon of Piatetski-Shapiro primes up to x corresponding to c, then |A|<< x(c)(1)/log x (log log log log x)(2-s/dc +epsilon).

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This work is supported by the National Natural Science Foundation of China (Grant Nos. 11871307 and 12031008) and the National Key Research and Development Program of China (Grant No. 2021YFA1000700). The second author was supported by UTUGS funding, working in the Academy of Finland Project No. 333707