Iterated sumset expansion in $\mathbb{F}_p^n$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917001454157824 |
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| author | Dhar, Manik Luo, Sammy |
| author_facet | Dhar, Manik Luo, Sammy |
| contents | Given a set $A \subseteq \mathbb{F}_p^n$, what conditions does one need to guarantee that iterated sumsets of the form $A+\cdots+A$ expand quickly (say, within $O(p)$ terms) to the whole space? When only the size of $A$ is known, such expansion results are only possible when $|A|>\frac{1}{p}|\mathbb{F}_p^n|$. However, heuristic considerations suggest that expansion should begin with much smaller sets under just mild ``nondegeneracy'' conditions. In this paper, we confirm this intuition by showing a sufficient algebraic condition for the asymmetric version of this problem: We have $A_1+\dots+A_m=\mathbb{F}_p^n$ as long as each $A_i$ is not contained in the zero set of any low degree polynomial ($\text{deg} = O(n)$ when $m=O(p)$). We close with a discussion of the behavior of random sets, as well as extensions of these results and connections with the Erdős-Ginzburg-Ziv problem. Our proofs make use of the shift operator polynomial method developed by the second author. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_08857 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Iterated sumset expansion in $\mathbb{F}_p^n$ Dhar, Manik Luo, Sammy Combinatorics 05D40 Given a set $A \subseteq \mathbb{F}_p^n$, what conditions does one need to guarantee that iterated sumsets of the form $A+\cdots+A$ expand quickly (say, within $O(p)$ terms) to the whole space? When only the size of $A$ is known, such expansion results are only possible when $|A|>\frac{1}{p}|\mathbb{F}_p^n|$. However, heuristic considerations suggest that expansion should begin with much smaller sets under just mild ``nondegeneracy'' conditions. In this paper, we confirm this intuition by showing a sufficient algebraic condition for the asymmetric version of this problem: We have $A_1+\dots+A_m=\mathbb{F}_p^n$ as long as each $A_i$ is not contained in the zero set of any low degree polynomial ($\text{deg} = O(n)$ when $m=O(p)$). We close with a discussion of the behavior of random sets, as well as extensions of these results and connections with the Erdős-Ginzburg-Ziv problem. Our proofs make use of the shift operator polynomial method developed by the second author. |
| title | Iterated sumset expansion in $\mathbb{F}_p^n$ |
| topic | Combinatorics 05D40 |
| url | https://arxiv.org/abs/2510.08857 |