Additive energy, Freiman's lemma, and an edge-isoperimetric problem on the boolean cube
Abstract
Let d ∈ ℕ and let A ⊆ ℝd be a finite, non-empty set not contained in any translate of a hyperplane. We show that there exists a subset S ⊆ A such that d ≪ |S| ≪ d and E(A,S) := |{(a1,a2,s1,s2)∈ A× A× S× S:a1+s1=a2+s2}|≪lvert Arvertlvert Srvert. This can be interpreted as a robust, analytic generalisation of the classical Freiman's lemma, and recovers Freiman's lemma up to a constant multiplicative factor. In the process of proving this result, we also show that for any integer 1 ≤ m ≤ d and any set A ⊂ ℝd with |A| ≤ binomdm, one has E(A, B) ≤ 4(m + 1/m) |A||B|, where B is the canonical basis of ℤd. This gives new estimates for multiplicative energies between finite sets of primes and arbitrary subsets of positive integers, improving a previous result of Shkredov. Using related ideas, we also prove new bounds for an edge-isoperimetric problem for certain graphs on the boolean cube, improving an earlier result of Kirshner and Samorodnitsky. We give two different proofs of the first result described in this abstract and three different proofs of the second one.
Related
- Double-arc rearrangement and the symmetric isoperimetric problem in Gauss space
- A comparison theorem for the planar strip isoperimetric problem
- The isoperimetric problem for four-dimensional parallelohedra
- The isoperimetric problem in the Riemannian Heisenberg $3$-space
- A complete solution to the reverse isoperimetric problem for λ-convex bodies
BibTeX
@misc{mudgal2026additive,
title = {{Additive energy, Freiman's lemma, and an edge-isoperimetric problem on the boolean cube}},
author = {Akshat Mudgal and Carl Schildkraut and Dmitrii Zakharov},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04644},
eprint = {2611.04644},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04644},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Additive energy, Freiman's lemma, and an edge-isoperimetric problem on the boolean cube
AU - Akshat Mudgal
AU - Carl Schildkraut
AU - Dmitrii Zakharov
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04644
UR - https://arxiv.org/abs/2611.04644
AB - Let d ∈ ℕ and let A ⊆ ℝ^d be a finite, non-empty set not contained in any translate of a hyperplane. We show that there exists a subset S ⊆ A such that d ≪ |S| ≪ d and E(A,S) := |{(a_1,a_2,s_1,s_2)∈ A× A× S× S:a_1+s_1=a_2+s_2}|≪lvert Arvertlvert Srvert. This can be interpreted as a robust, analytic generalisation of the classical Freiman's lemma, and recovers Freiman's lemma up to a constant multiplicative factor. In the process of proving this result, we also show that for any integer 1 ≤ m ≤ d and any set A ⊂ ℝ^d with |A| ≤ binomdm, one has E(A, B) ≤ 4(m + 1/m) |A||B|, where B is the canonical basis of ℤ^d. This gives new estimates for multiplicative energies between finite sets of primes and arbitrary subsets of positive integers, improving a previous result of Shkredov. Using related ideas, we also prove new bounds for an edge-isoperimetric problem for certain graphs on the boolean cube, improving an earlier result of Kirshner and Samorodnitsky. We give two different proofs of the first result described in this abstract and three different proofs of the second one.
ER -