An m2.943 Bohnenblust--Hille Bound on the Boolean Cube
Abstract
Let qm=2m/(m+1) and put β0=3/2+1/(log 2)=2.9426950408…, where log is the natural logarithm. We give a proof scheme showing that, for every ε>0, there is Cε<∞ such that every complex-valued function f:{−1,1}n→C of Fourier degree at most m satisfies (∑A⊆[n]abswh f(A)qm)1/qm ≤ Cε mβ0+εnormf∞. The improvement over the m9 estimate of the earlier draft has two ingredients. The first three Fourier levels are estimated at the scales m, m3/2, and m5/3. These losses are encoded in the weight Mμr/rrBmin{r,LM}8, LMasympB log (M+1), with μr=⌈3r/2⌉ for r≥2. A parity-compatible central window handles r<LM, while a shrinking balanced window handles r≥ LMasympBlog(M+1), no parity is used in this regime. The leading high-degree loss--gain factor is ee 2−B; careful uniform bounds close the bootstrap for every B>1/log2. The formula for μr is explained below.
Related
- Polynomial growth of Bohnenblust--Hille constants on the Hamming cube
- Support-Sensitive Bohnenblust-Hille Inequalities and Local Invariants on Hamming Schemes
- Asymptotic Contractivity of Bohnenblust-Hille Constants with Bounded Monomial Support
- Polynomial Bohnenblust--Hille bounds for product of cyclic groups
- Polynomial growth of complex polynomial Bohnenblust--Hille constants
BibTeX
@misc{slote2026m2943,
title = {{An $m^2.943$ Bohnenblust--Hille Bound on the Boolean Cube}},
author = {Joseph Slote and Chun-Kai Tseng and Alexander Volberg},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21144},
eprint = {2609.21144},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21144},
}
RIS
TY - UNPB
TI - An m^2.943 Bohnenblust--Hille Bound on the Boolean Cube
AU - Joseph Slote
AU - Chun-Kai Tseng
AU - Alexander Volberg
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.21144
UR - https://arxiv.org/abs/2609.21144
AB - Let q_m=2m/(m+1) and put β_0=3/2+1/(log 2)=2.9426950408…, where log is the natural logarithm. We give a proof scheme showing that, for every ε>0, there is C_ε<∞ such that every complex-valued function f:{−1,1}^n→C of Fourier degree at most m satisfies (∑_A⊆[n]abswh f(A)^q_m)^1/q_m ≤ C_ε m^β_0+εnormf_∞. The improvement over the m^9 estimate of the earlier draft has two ingredients. The first three Fourier levels are estimated at the scales m, m^3/2, and m^5/3. These losses are encoded in the weight M^μ_r/rr^Bmin{r,L_M}^8, L_Masymp_B log (M+1), with μ_r=⌈3r/2⌉ for r≥2. A parity-compatible central window handles r<L_M, while a shrinking balanced window handles r≥ L_Masymp_Blog(M+1), no parity is used in this regime. The leading high-degree loss--gain factor is ee 2^−B; careful uniform bounds close the bootstrap for every B>1/log2. The formula for μ_r is explained below.
ER -