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

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 $N^{6\mathrm{/}25+\epsilon }\le H\le N^{1\mathrm{/}4-\epsilon }$ and improves the previously known admissible lower bound for $H$.