A dot-product bound from separate growth-minimizing bases
Abstract
For every finite P⊂ℝ2 we prove |{p· q: p,q∈ P}|≫ |P|199/295, with an absolute constant and no logarithmic loss, where 199/295 = 2/3+7/885. This improves the bound 2/3+7/1425 of Kokkinos (arXiv:2502.12727), which in turn had improved the first superthreshold bound of Hanson, Roche-Newton, and Senger. The main new ingredient is the inequality |F(2)G(2)/(F(2)G)| ≤ |AB|(|AF|/|A|)4(|BG|/|B|)3: each radial profile retains its own Petridis growth-minimizing subset, and the product of the two subsets serves as a common base for both growth operators; a prime-power construction shows this is sharp under its hypotheses. The second ingredient is a weighted-median replacement for the dyadic selection in the squeezing argument of Roche-Newton and Wong, upgrading their seven-factor expander to a logarithm-free form; an appendix gives the complete proof from the Solymosi-Zahl incidence theorem. We complement the lower bound with a third-moment structure theorem for near-extremal configurations and a construction showing that the pinned problem admits no linear lower bound: maxp∈ P|p· P| = O(|P|/√(log |P|)) is attainable. The algebraic core, the median lemma, the surface identities, and the pinned construction are formally verified in Lean 4 and included as ancillary files.
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BibTeX
@misc{lu2026dotproduct,
title = {{A dot-product bound from separate growth-minimizing bases}},
author = {Zhipeng Lu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06730},
eprint = {2609.06730},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06730},
}
RIS
TY - UNPB
TI - A dot-product bound from separate growth-minimizing bases
AU - Zhipeng Lu
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06730
UR - https://arxiv.org/abs/2609.06730
AB - For every finite P⊂ℝ^2 we prove |{p· q: p,q∈ P}|≫ |P|^199/295, with an absolute constant and no logarithmic loss, where 199/295 = 2/3+7/885. This improves the bound 2/3+7/1425 of Kokkinos (arXiv:2502.12727), which in turn had improved the first superthreshold bound of Hanson, Roche-Newton, and Senger. The main new ingredient is the inequality |F^(2)G^(2)/(F^(2)G)| ≤ |AB|(|AF|/|A|)^4(|BG|/|B|)^3: each radial profile retains its own Petridis growth-minimizing subset, and the product of the two subsets serves as a common base for both growth operators; a prime-power construction shows this is sharp under its hypotheses. The second ingredient is a weighted-median replacement for the dyadic selection in the squeezing argument of Roche-Newton and Wong, upgrading their seven-factor expander to a logarithm-free form; an appendix gives the complete proof from the Solymosi-Zahl incidence theorem. We complement the lower bound with a third-moment structure theorem for near-extremal configurations and a construction showing that the pinned problem admits no linear lower bound: max_p∈ P|p· P| = O(|P|/√(log |P|)) is attainable. The algebraic core, the median lemma, the surface identities, and the pinned construction are formally verified in Lean 4 and included as ancillary files.
ER -