| Autor: |
Elsholtz, Christian, Proske, Laura, Sauermann, Lisa |
| Rok vydání: |
2024 |
| Předmět: |
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| Druh dokumentu: |
Working Paper |
| Popis: |
We prove new lower bounds on the maximum size of sets $A\subseteq \mathbb{F}_p^n$ or $A\subseteq \mathbb{Z}_m^n$ not containing three-term arithmetic progressions (consisting of three distinct points). More specifically, we prove that for any fixed integer $m\ge 2$ and sufficiently large $n$ (in terms of $m$), there exists a three-term progression free subset $A\subseteq \mathbb{Z}_m^n$ of size $|A|\ge (cm)^n$ for some absolute constant $c>1/2$. Such a bound for $c=1/2$ can be obtained with a classical construction of Salem and Spencer from 1942, and improving upon this value of $1/2$ has been a well-known open problem (our proof gives $c= 0.54$). Our construction relies on finding a subset $S\subset \mathbb{Z}_m^2$ of size at least $(7/24)m^2$ with a certain type of reducibility property. This property allows us to ``lift'' $S$ to a three-term progression free subset of $\mathbb{Z}_m^n$ for large $n$ (even though the original set $S\subset \mathbb{Z}_m^2$ does contain three-term arithmetic progressions). |
| Databáze: |
arXiv |
| Externí odkaz: |
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