PreprintCC BY 4.0arXiv
Trilinear Kloosterman fractions II: subdyadic intervals and nearly balanced convolutions
Abstract
This paper broadens the range on which Fouvry and Radziwiłł's results on nearly balanced convolutions apply. In particular, let αm and βn be sequences supported on m∼ M and n∼ N where βn is equidistributed for small moduli, and let Q=X1/2+ε. We find that ∑q∼ Q|mathop∑∑substackn∼ N,m∼ M mn≡ apmod qαmβn−1/ϕ(q)mathop∑∑substackn∼ N,m∼ M (mn,q)=1αmβn|≪ X/(logA X) if N=X1/2+δ and M=X1/2−δ with 0<δ<1/68, which improves Fouvry and Radziwiłł's 0<δ<1/112. To prove this, we sharpen Bettin and Chandee's famous result on trilinear forms with Kloosterman fractions in the case where some of the sums are over subdyadic intervals.
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BibTeX
@misc{wright2026trilinear,
title = {{Trilinear Kloosterman fractions II: subdyadic intervals and nearly balanced convolutions}},
author = {Thomas Wright},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27732},
eprint = {2608.27732},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27732},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Trilinear Kloosterman fractions II: subdyadic intervals and nearly balanced convolutions
AU - Thomas Wright
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2608.27732
UR - https://arxiv.org/abs/2608.27732
AB - This paper broadens the range on which Fouvry and Radziwiłł's results on nearly balanced convolutions apply. In particular, let α_m and β_n be sequences supported on m∼ M and n∼ N where β_n is equidistributed for small moduli, and let Q=X^1/2+ε. We find that ∑_q∼ Q|mathop∑∑_substackn∼ N,m∼ M mn≡ apmod qα_mβ_n−1/ϕ(q)mathop∑∑_substackn∼ N,m∼ M (mn,q)=1α_mβ_n|≪ X/(log^A X) if N=X^1/2+δ and M=X^1/2−δ with 0<δ<1/68, which improves Fouvry and Radziwiłł's 0<δ<1/112. To prove this, we sharpen Bettin and Chandee's famous result on trilinear forms with Kloosterman fractions in the case where some of the sums are over subdyadic intervals.
ER -