PreprintarXiv licencearXiv
The main conjecture for a Vinogradov subsystem
Abstract
When k≥ 2 and s>0, we establish that for each ε>0, one has ∫[0,1)k−1biggl| ∑1≤ x≤ Xe(α2x2+… +αk xk)biggr|2s dα≪ Xs+ε+X2s−(k2+k−2)/2, confirming the main conjecture for a new family of exponential sums beyond those of Vinogradov (translation-dilation invariant) type. Our methods are based on very recent work of the author employing the nested efficient congruencing method. Consequently, analogues of our results hold also in the setting of number fields and function fields.
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BibTeX
@misc{wooley2026main,
title = {{The main conjecture for a Vinogradov subsystem}},
author = {Trevor D. Wooley},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05515},
eprint = {2610.05515},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05515},
}
RIS
TY - UNPB
TI - The main conjecture for a Vinogradov subsystem
AU - Trevor D. Wooley
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05515
UR - https://arxiv.org/abs/2610.05515
AB - When k≥ 2 and s>0, we establish that for each ε>0, one has ∫_[0,1)^k−1biggl| ∑_1≤ x≤ Xe(α_2x^2+… +α_k x^k)biggr|^2s dα≪ X^s+ε+X^2s−(k^2+k−2)/2, confirming the main conjecture for a new family of exponential sums beyond those of Vinogradov (translation-dilation invariant) type. Our methods are based on very recent work of the author employing the nested efficient congruencing method. Consequently, analogues of our results hold also in the setting of number fields and function fields.
ER -