Majorization and additive tuples in ℤ_2^n

Abstract

Majorization is a fundamental tool for comparing how "spread out" the entries of two vectors are. Key majorization results were obtained for the integers by Hardy, Littlewood and Pólya and for ℤ_p by Lev. In this paper, we establish a powerful majorization theorem in ℤ₂ⁿ that is an analogue of Lev's result in ℤ_p. Our proof is based on a novel use of a compression argument that optimizes certain sums of additive representation counts. Our majorization theorem has several applications in additive combinatorics. We resolve the question: given subsets A₁,…,A_k⊆ℤ₂ⁿ of prescribed sizes, when is the function r_k(A₁,…,A_k), which counts the number of additive k-tuples in A₁×⋯× A_k, maximized? We establish the corresponding minimization result and a characterization of all extremizers for r_k(A,…,A) when k is odd. When k=3, this quantity is the number of Schur triples in A; as a special case, we recover a theorem of Samotij and Sudakov on Schur triples. We also obtain a convexity inequality and use it to prove an analogue of Pollard's theorem that strengthens and extends a well-known result of Bollobás and Leader in the ℤ₂ⁿ setting.

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BibTeX RIS

BibTeX

@misc{huczynska2026majorization,
  title = {{Majorization and additive tuples in $\mathbbZ_2^n$}},
  author = {Sophie Huczynska and Firdavs Rakhmonov and Chi Hoi Yip},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07855},
  eprint = {2609.07855},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07855},
}

RIS

TY  - UNPB
TI  - Majorization and additive tuples in ℤ_2^n
AU  - Sophie Huczynska
AU  - Firdavs Rakhmonov
AU  - Chi Hoi Yip
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.07855
UR  - https://arxiv.org/abs/2609.07855
AB  - Majorization is a fundamental tool for comparing how "spread out" the entries of two vectors are. Key majorization results were obtained for the integers by Hardy, Littlewood and Pólya and for ℤ_p by Lev. In this paper, we establish a powerful majorization theorem in ℤ_2^n that is an analogue of Lev's result in ℤ_p. Our proof is based on a novel use of a compression argument that optimizes certain sums of additive representation counts. Our majorization theorem has several applications in additive combinatorics. We resolve the question: given subsets A_1,…,A_k⊆ℤ_2^n of prescribed sizes, when is the function r_k(A_1,…,A_k), which counts the number of additive k-tuples in A_1×⋯× A_k, maximized? We establish the corresponding minimization result and a characterization of all extremizers for r_k(A,…,A) when k is odd. When k=3, this quantity is the number of Schur triples in A; as a special case, we recover a theorem of Samotij and Sudakov on Schur triples. We also obtain a convexity inequality and use it to prove an analogue of Pollard's theorem that strengthens and extends a well-known result of Bollobás and Leader in the ℤ_2^n setting.
ER  -

Majorization and additive tuples in ℤ₂ⁿ

Abstract

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