Abstract
Let N be a sufficiently large number, \(\mathfrak{A}\) and \(\mathfrak{B}\) be subsets of \(\{N+1, \ldots , 2N\}\). We prove that if \(1<c<\frac{6}{5}\), \(|\mathfrak{A}|\, |\mathfrak{B}|\gg N^{2-2\delta}\) and \(\delta>0\) is sufficiently small, then the equation
is solvable, which improves the result of Rivat and Sárközy [14]. We also investigate the solvability of the equation
where Pk denotes an almost-prime with at most k prime factors and c0 is a fixed real number depends on k.
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The authors express the most sincere gratitude to the referee for patience and time in refereeing this paper.
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This research is supported by National Natural Science Foundation of China (Grant No. 11971476, 11901566).
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Zhai, WG., Zhao, YT. On a Piatetski-Shapiro analog problem over almost-primes. Acta Math. Hungar. 170, 616–632 (2023). https://doi.org/10.1007/s10474-023-01371-1
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DOI: https://doi.org/10.1007/s10474-023-01371-1