Sharp lower bounds for sumsets in hypercubes
Abstract
We prove a sharp lower bound for the cardinality of sumsets of subsets of ℤd confined to a hypercube, resolving in strong form a conjecture that was made explicit by Becker, Ivanisvili, Krachun and Madrid and had circulated in the folklore of the field for some time. Specifically, for sets Aj⊆ {0,1,2,…,m}d we show that |A1+…+An| ≥ (|A1|⋯|An|)1/p, p=(nlog(m+1))/(log(nm+1)), with the exponent best possible. The only previously known sharp cases were Aj⊆ {0,1}d, for all n≥1, and Aj⊆ {0,1,2}d for n=2. We also prove a sharp inequality in the case when Aj⊆{0,1,…,mj}d for different mj. We obtain the above inequality as a corollary of a stronger result on sup-convolution of functions on ℤd, whose proof is based on a novel mixed volume representation of a lattice path norm, together with a sharp one-dimensional functional inequality.
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BibTeX
@misc{goncalves2026sharp,
title = {{Sharp lower bounds for sumsets in hypercubes}},
author = {Felipe Gonçalves and Danylo Radchenko},
howpublished = {HAL},
year = {2026},
month = jul,
url = {https://hal.science/hal-05851856v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp lower bounds for sumsets in hypercubes
AU - Felipe Gonçalves
AU - Danylo Radchenko
PY - 2026
DA - 2026/07/01
PB - HAL
UR - https://hal.science/hal-05851856v1
AB - We prove a sharp lower bound for the cardinality of sumsets of subsets of ℤ^d confined to a hypercube, resolving in strong form a conjecture that was made explicit by Becker, Ivanisvili, Krachun and Madrid and had circulated in the folklore of the field for some time. Specifically, for sets A_j⊆ {0,1,2,…,m}^d we show that |A_1+…+A_n| ≥ (|A_1|⋯|A_n|)^1/p, p=(nlog(m+1))/(log(nm+1)), with the exponent best possible. The only previously known sharp cases were A_j⊆ {0,1}^d, for all n≥1, and A_j⊆ {0,1,2}^d for n=2. We also prove a sharp inequality in the case when A_j⊆{0,1,…,m_j}^d for different m_j. We obtain the above inequality as a corollary of a stronger result on sup-convolution of functions on ℤ^d, whose proof is based on a novel mixed volume representation of a lattice path norm, together with a sharp one-dimensional functional inequality.
ER -