关于等差数列中\(k\)次幂数量的评论
A comment on the number of $k$-th powers inside arithmetic progressions
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中文总结 AI 辅助
研究等差数列中\(k\)次幂数量，布尔甘和德梅特的论证在步长条件变化时仍有效，且给出足够大\(N\)时前\(N\)项中\(k\)次幂数量精确界，虽可能已知但为文献添内容。
中文摘要 AI 辅助
在文献\(\cite{BD}\)中，布尔甘（Bourgain）和德梅特（Demeter）找到了任意等差数列中\(k\)次幂数量的精确上界，该等差数列的步长具有\(O(1)\)个因数。我们不难发现，如果步长相对于数列长度增长不过快，相同的论证仍然有效。此外，对于足够大的\(N\)，我们给出了前\(N\)项中\(k\)次幂数量的精确界。这两个结果可能已为人知，但我们仍为文献增添了内容。
英文摘要
In \cite{BD} Bourgain and Demeter found sharp upper bounds for the number of $k$-th powers inside arbitrary arithmetic progressions whose step has $O(1)$ many divisors. We make the easy observation that the same arguments are still valid if the step does not grow too rapidly in relation to the length of the progression. Furthermore, we give sharp bounds for the number of $k$-th powers among the first $N$ terms for $N$ large enough. Both results should be known. Nevertheless, we add to the literature.
