Ideal-Smooth Sets and an Explicit Upper Bound for the Real Sum-Product Exponent
Abstract
We construct arbitrarily large finite sets A of real algebraic integers such that both |A+A| and |AA| are at most |A|1.95835. The construction combines truncated ideal-smooth S-unit fibres with a coprime additive factor. A tensor-product rank argument, using a two-dimensional local feature at each selected prime ideal, controls the loss in the additive factor, while a direct estimate for an outer parallel body improves the sumset packing bound. The arithmetic input is an unramified pro-$2$ tower over a known degree-ten field, with simultaneous Frobenius cuts controlling the small prime ideals. We use unconditional Tsfasman--Vlăduţ inequalities for the joint class-number--regulator cost. All finite numerical comparisons entering the exponent are certified by outward rational interval arithmetic. No unproved hypothesis is used.
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BibTeX
@misc{zhang2026idealsmooth,
title = {{Ideal-Smooth Sets and an Explicit Upper Bound for the Real Sum-Product Exponent}},
author = {Kaiqiang Zhang and Yuanguo Zeng and Yuyu Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05725},
eprint = {2610.05725},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05725},
}
RIS
TY - UNPB
TI - Ideal-Smooth Sets and an Explicit Upper Bound for the Real Sum-Product Exponent
AU - Kaiqiang Zhang
AU - Yuanguo Zeng
AU - Yuyu Wang
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.05725
UR - https://arxiv.org/abs/2610.05725
AB - We construct arbitrarily large finite sets A of real algebraic integers such that both |A+A| and |AA| are at most |A|^1.95835. The construction combines truncated ideal-smooth S-unit fibres with a coprime additive factor. A tensor-product rank argument, using a two-dimensional local feature at each selected prime ideal, controls the loss in the additive factor, while a direct estimate for an outer parallel body improves the sumset packing bound. The arithmetic input is an unramified pro-$2$ tower over a known degree-ten field, with simultaneous Frobenius cuts controlling the small prime ideals. We use unconditional Tsfasman--Vlăduţ inequalities for the joint class-number--regulator cost. All finite numerical comparisons entering the exponent are certified by outward rational interval arithmetic. No unproved hypothesis is used.
ER -