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تسجيل مستخدم جديدSimple proofs for Furstenberg sets over finite fields
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A $(k,m)$-Furstenberg set $S subset mathbb{F}_q^n$ over a finite field is a set that has at least $m$ points in common with a $k$-flat in every direction. The question of determining the smallest size of such sets is a natural generalization of the finite field Kakeya problem. The only previously known bound for these sets is due to Ellenberg-Erman and requires sophisticated machinery from algebraic geometry. In this work we give new, completely elementary and simple, proofs which significantly improve the known bounds. Our main result relies on an equivalent formulation of the problem using the notion of min-entropy, which could be of independent interest.
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A subset $S subset mathbb{F}_q^n$, where $mathbb{F}_q$ is a finite field, is called $(k,m)$-Furstenberg if it has $m$ common points with a $k$-flat in each direction. That is, any $k$-dimensional subspace of $mathbb{F}_q^n$ can be translated so that
it intersects $S$ in at least $m$ points. Using sophisticated scheme-theoretic machinery, Ellenberg and Erman proved that $(k,m)$-Furstenberg sets must have size at least $C_{n,k}m^{n/k}$ with a constant $C_{n,k}$ depending only $n$ and $k$. In this work we follow the overall proof strategy of Ellenberg-Erman, replacing the scheme-theoretic language with more elementary machinery. In addition to presenting the proof in a self-contained and accessible form, we are also able to improve the constant $C_{n,k}$ by modifying certain key parts of the argument.
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