Geodesic
Effective Bounds for Restricted $3$-Arithmetic Progressions in $\mathbb{F}_p^n$
Discrete analysis (2024) Cet article a éte moissonné depuis la source Scholastica

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For a prime $p$, a restricted arithmetic progression in $\mathbb{F}_p^n$ is a triplet of vectors $x, x+a, x+2a$ in which the common difference $a$ is a non-zero element from $\{0,1,2\}^n$. What is the size of the largest $A\subseteq \mathbb{F}_p^n$ that is free of restricted arithmetic progressions? We show that the density of any such a set is at most $\frac{C}{(\log\log\log n)^c}$, where $c,C>0$ depend only on $p$, giving the first reasonable bounds for the density of such sets. Previously, the best known bound was $O(1/\log^{*} n)$, which follows from the density Hales-Jewett theorem.
Publié le : 
@article{DAS_2024_a5,
     author = {Amey Bhangale and Subhash Khot and Dor Minzer},
     title = {Effective {Bounds} for {Restricted} $3${-Arithmetic} {Progressions} in $\mathbb{F}_p^n$},
     journal = {Discrete analysis},
     year = {2024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DAS_2024_a5/}
}
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AU  - Amey Bhangale
AU  - Subhash Khot
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TI  - Effective Bounds for Restricted $3$-Arithmetic Progressions in $\mathbb{F}_p^n$
JO  - Discrete analysis
PY  - 2024
UR  - http://geodesic.mathdoc.fr/item/DAS_2024_a5/
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ID  - DAS_2024_a5
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%F DAS_2024_a5
Amey Bhangale; Subhash Khot; Dor Minzer. Effective Bounds for Restricted $3$-Arithmetic Progressions in $\mathbb{F}_p^n$. Discrete analysis (2024). http://geodesic.mathdoc.fr/item/DAS_2024_a5/