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Sharp mean value estimates in the absence of translation-dilation invariance
Abstract
We establish sharp mean value estimates for various exponential sums lacking translation-dilation invariance properties. For example, suppose that k≥ 1 and s>0. Then, for each ε>0, one has ∫[0,1)kbiggl| ∑1≤ h≤ X∑1≤ z≤ Xe(hzα1+… +hzkαk)biggr|2s dα≪ Xε( Xs+X2s−k(k+3)/2), confirming the main conjecture for a new family of exponential sums. Our methods are based on the application of very recent work of the author employing the nested efficient congruencing method.
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BibTeX
@misc{wooley2026sharp,
title = {{Sharp mean value estimates in the absence of translation-dilation invariance}},
author = {Trevor D. Wooley},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05525},
eprint = {2610.05525},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05525},
}
RIS
TY - UNPB
TI - Sharp mean value estimates in the absence of translation-dilation invariance
AU - Trevor D. Wooley
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05525
UR - https://arxiv.org/abs/2610.05525
AB - We establish sharp mean value estimates for various exponential sums lacking translation-dilation invariance properties. For example, suppose that k≥ 1 and s>0. Then, for each ε>0, one has ∫_[0,1)^kbiggl| ∑_1≤ h≤ X∑_1≤ z≤ Xe(hzα_1+… +hz^kα_k)biggr|^2s dα≪ X^ε( X^s+X^2s−k(k+3)/2), confirming the main conjecture for a new family of exponential sums. Our methods are based on the application of very recent work of the author employing the nested efficient congruencing method.
ER -