PreprintCC BY-NC-ND 4.0arXiv
Intermediate-Range Estimates for Short Weyl Sums and Waring's Problem with Almost Proportional Summands
Abstract
We obtain a uniform pointwise estimate for short Weyl sums of degree n≥3 in the intermediate rational-approximation range 1/(qxn−2y)≪|λ| ≪1/(qyn−1). This range arises from a second application of Dirichlet's rational approximation theorem in estimating the residual integral that occurs in the derivation of an asymptotic formula for Waring's problem with almost proportional summands. For r=2n+1 and fixed positive numbers μ1,…,μr satisfying μ1+⋯+μr=1, we derive an asymptotic formula for the number of representations x1n+⋯+xrn=N, |xin−μiN|≤ H, 1≤ i ≤ r, valid for N1−θ(n,r)+ε≤ H ≤ N/(ln N), θ(n,r)= 2/(n((r−1)(n−1)+2)). The resulting admissible lower bound for H improves the previously known bound for every n≥3.
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BibTeX
@misc{mirzoabdughafurov2026intermediaterange,
title = {{Intermediate-Range Estimates for Short Weyl Sums and Waring's Problem with Almost Proportional Summands}},
author = {Karimjon Ibrohimjonovich Mirzoabdughafurov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.25787},
eprint = {2608.25787},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.25787},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Intermediate-Range Estimates for Short Weyl Sums and Waring's Problem with Almost Proportional Summands
AU - Karimjon Ibrohimjonovich Mirzoabdughafurov
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2608.25787
UR - https://arxiv.org/abs/2608.25787
AB - We obtain a uniform pointwise estimate for short Weyl sums of degree n≥3 in the intermediate rational-approximation range 1/(qx^n−2y)≪|λ| ≪1/(qy^n−1). This range arises from a second application of Dirichlet's rational approximation theorem in estimating the residual integral that occurs in the derivation of an asymptotic formula for Waring's problem with almost proportional summands. For r=2^n+1 and fixed positive numbers μ_1,…,μ_r satisfying μ_1+⋯+μ_r=1, we derive an asymptotic formula for the number of representations x_1^n+⋯+x_r^n=N, |x_i^n−μ_iN|≤ H, 1≤ i ≤ r, valid for N^1−θ(n,r)+ε≤ H ≤ N/(ln N), θ(n,r)= 2/(n((r−1)(n−1)+2)). The resulting admissible lower bound for H improves the previously known bound for every n≥3.
ER -