A counterexample to a conjecture of Sárközy on sums and products modulo a prime
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918421126447104 |
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| author | Tang, Quanyu |
| author_facet | Tang, Quanyu |
| contents | Let $p$ be a prime and, for $A\subseteq \mathbb F_p$, define $A^\ast=(A+A)\cup(AA)$. Sárközy conjectured that there exist constants $c>0$ and $p_0$ such that, for every prime $p>p_0$, every set $A\subseteq \mathbb F_p$ with $|A|>\left(\frac12-c\right)p$ satisfies $\mathbb F_p^\times\subseteq A^\ast$. We disprove this conjecture: for every odd prime $p\ge 5$, there exists a set $A\subseteq \mathbb F_p$ with $|A|=\frac{p-1}{2}$ such that $1\notin A^\ast$. Thus no positive constant $c$ can satisfy Sárközy's conjecture. Conversely, if $|A|>\frac{p}{2}$, then $A+A=\mathbb F_p$. Therefore the sharp threshold is exactly $\frac12$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29992 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A counterexample to a conjecture of Sárközy on sums and products modulo a prime Tang, Quanyu Number Theory 11B75, 05C69 Let $p$ be a prime and, for $A\subseteq \mathbb F_p$, define $A^\ast=(A+A)\cup(AA)$. Sárközy conjectured that there exist constants $c>0$ and $p_0$ such that, for every prime $p>p_0$, every set $A\subseteq \mathbb F_p$ with $|A|>\left(\frac12-c\right)p$ satisfies $\mathbb F_p^\times\subseteq A^\ast$. We disprove this conjecture: for every odd prime $p\ge 5$, there exists a set $A\subseteq \mathbb F_p$ with $|A|=\frac{p-1}{2}$ such that $1\notin A^\ast$. Thus no positive constant $c$ can satisfy Sárközy's conjecture. Conversely, if $|A|>\frac{p}{2}$, then $A+A=\mathbb F_p$. Therefore the sharp threshold is exactly $\frac12$. |
| title | A counterexample to a conjecture of Sárközy on sums and products modulo a prime |
| topic | Number Theory 11B75, 05C69 |
| url | https://arxiv.org/abs/2603.29992 |