On Graham's rearrangement conjecture over $\mathbb{F}_2^n$
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912553375891456 |
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| author | Bedert, Benjamin Bucić, Matija Kravitz, Noah Montgomery, Richard Müyesser, Alp |
| author_facet | Bedert, Benjamin Bucić, Matija Kravitz, Noah Montgomery, Richard Müyesser, Alp |
| contents | A sequence $s_1,s_2,\ldots, s_k$ of elements of a group $G$ is called a valid ordering if the partial products $s_1, s_1 s_2, \ldots, s_1\cdots s_k$ are all distinct. A long-standing problem in combinatorial group theory asks whether, for a given group $G$, every subset $S \subseteq G\setminus \{\mathrm{id}\}$ admits a valid ordering; the instance of the additive group $\mathbb{F}_p$ is the content of a well-known 1971 conjecture of Graham. Most partial progress to date has concerned the edge cases where either $S$ or $G \setminus S$ is quite small. Our main result is an essentially complete resolution of the problem for $G=\mathbb{F}_2^n$: we show that there is an absolute constant $C>0$ such that every subset $S\subseteq \mathbb{F}_2^n \setminus \{0\}$ of size at least $C$ admits a valid ordering. Our proof combines techniques from additive and probabilistic combinatorics, including the Freiman--Ruzsa theorem and the absorption method.
Along the way, we also solve the general problem for moderately large subsets: there is a constant $c>0$ such that for every group $G$ (not necessarily abelian), every subset $S \subseteq G\setminus \{\mathrm{id}\}$ of size at least $|G|^{1-c}$ admits a valid ordering. Previous work in this direction concerned only sets of size at least $(1-o(1))|G|$. A main ingredient in our proof is a structural result, similar in spirit to the Arithmetic Regularity Lemma, showing that every Cayley graph can be efficiently decomposed into mildly quasirandom components. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18254 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Graham's rearrangement conjecture over $\mathbb{F}_2^n$ Bedert, Benjamin Bucić, Matija Kravitz, Noah Montgomery, Richard Müyesser, Alp Combinatorics Group Theory Number Theory 05C35, 05D40, 20K01 A sequence $s_1,s_2,\ldots, s_k$ of elements of a group $G$ is called a valid ordering if the partial products $s_1, s_1 s_2, \ldots, s_1\cdots s_k$ are all distinct. A long-standing problem in combinatorial group theory asks whether, for a given group $G$, every subset $S \subseteq G\setminus \{\mathrm{id}\}$ admits a valid ordering; the instance of the additive group $\mathbb{F}_p$ is the content of a well-known 1971 conjecture of Graham. Most partial progress to date has concerned the edge cases where either $S$ or $G \setminus S$ is quite small. Our main result is an essentially complete resolution of the problem for $G=\mathbb{F}_2^n$: we show that there is an absolute constant $C>0$ such that every subset $S\subseteq \mathbb{F}_2^n \setminus \{0\}$ of size at least $C$ admits a valid ordering. Our proof combines techniques from additive and probabilistic combinatorics, including the Freiman--Ruzsa theorem and the absorption method. Along the way, we also solve the general problem for moderately large subsets: there is a constant $c>0$ such that for every group $G$ (not necessarily abelian), every subset $S \subseteq G\setminus \{\mathrm{id}\}$ of size at least $|G|^{1-c}$ admits a valid ordering. Previous work in this direction concerned only sets of size at least $(1-o(1))|G|$. A main ingredient in our proof is a structural result, similar in spirit to the Arithmetic Regularity Lemma, showing that every Cayley graph can be efficiently decomposed into mildly quasirandom components. |
| title | On Graham's rearrangement conjecture over $\mathbb{F}_2^n$ |
| topic | Combinatorics Group Theory Number Theory 05C35, 05D40, 20K01 |
| url | https://arxiv.org/abs/2508.18254 |