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A Refined Sum-Product Estimate via Higher Energies
Abstract
Let A⊂ℝ be a finite set. Combining the multiplicative slope estimate of Rudnev--Stevens, Cushman's higher-energy regularization, Shakan's d+--d× decomposition, and Solymosi's classical sum--product estimate, we prove |AA|204|A+A|301≳ |A|675, where ≳ suppresses a fixed polylogarithmic factor in |A|. Consequently, for every ε>0, max{|A+A|,|AA|}≫ε |A|135/101−ε. The proof is organized around two intermediate estimates. For every nonempty finite set B⊂ℝ>0, d×(B)|BB|12|B+B|16≳ |B|38, whereas for every nonempty finite set U⊂ℝ, d+(U)17|U+U|29≳ |U|46.
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BibTeX
@misc{zhang2026refined,
title = {{A Refined Sum-Product Estimate via Higher Energies}},
author = {Kaiqiang Zhang and Yuyu Wang and Yuanguo Zeng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25711},
eprint = {2609.25711},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25711},
}
RIS
TY - UNPB
TI - A Refined Sum-Product Estimate via Higher Energies
AU - Kaiqiang Zhang
AU - Yuyu Wang
AU - Yuanguo Zeng
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25711
UR - https://arxiv.org/abs/2609.25711
AB - Let A⊂ℝ be a finite set. Combining the multiplicative slope estimate of Rudnev--Stevens, Cushman's higher-energy regularization, Shakan's d^+--d^× decomposition, and Solymosi's classical sum--product estimate, we prove |AA|^204|A+A|^301≳ |A|^675, where ≳ suppresses a fixed polylogarithmic factor in |A|. Consequently, for every ε>0, max{|A+A|,|AA|}≫_ε |A|^135/101−ε. The proof is organized around two intermediate estimates. For every nonempty finite set B⊂ℝ_>0, d^×(B)|BB|^12|B+B|^16≳ |B|^38, whereas for every nonempty finite set U⊂ℝ, d^+(U)^17|U+U|^29≳ |U|^46.
ER -