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arXiv · 2607.25671

Short Quartic Exponential Sums in an Intermediate Range and Waring's Problem for Fourth Powers with Almost Equal Summands

Karimjon Ibrohimjonovich Mirzoabdughafurov
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Abstract

This paper is devoted to short quartic Weyl sums in an intermediate range of rational approximations and to their application to Waring's problem with almost equal summands. We obtain a pointwise estimate for such sums and use it, together with the Hardy--Littlewood circle method, to establish an asymptotic formula for the number of representations of a sufficiently large natural number as a sum of seventeen fourth powers of integers lying in a short interval about a common centre. The asymptotic formula is valid for $N^{6/25+\varepsilon}\leq H\leq N^{1/4-\varepsilon}$ and improves the previously known admissible lower bound for $H$.

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Karimjon Ibrohimjonovich Mirzoabdughafurov. 2026-07-28. Short Quartic Exponential Sums in an Intermediate Range and Waring's Problem for Fourth Powers with Almost Equal Summands. https://arxiv.org/abs/2607.25671

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