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March 2020 Congruence classes of large configurations in vector spaces over finite fields
Alex McDonald
Funct. Approx. Comment. Math. 62(1): 131-141 (March 2020). DOI: 10.7169/facm/1814

Abstract

In [1], Bennett, Hart, Iosevich, Pakianathan, and Rudnev found an exponent $s<d$ such that any set $E\subset \mathbb{F}_q^d$ with $|E|\gtrsim q^s$ determines $\gtrsim q^{\binom{k+1}{2}}$ congruence classes of $(k+1)$-point configurations for $k\leq d$. Because congruence classes can be identified with tuples of distances between distinct points when $k\leq d$, and because there are $\binom{k+1}{2}$ such pairs, this means any such $E$ determines a positive proportion of all congruence classes. In the $k>d$ case, fixing all pairs of distnaces leads to an overdetermined system, so $q^{\binom{k+1}{2}}$ is no longer the correct number of congruence classes. We determine the correct number, and prove that $|E|\gtrsim q^s$ still determines a positive proportion of all congruence classes, for the same $s$ as in the $k\leq d$ case.

Citation

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Alex McDonald. "Congruence classes of large configurations in vector spaces over finite fields." Funct. Approx. Comment. Math. 62 (1) 131 - 141, March 2020. https://doi.org/10.7169/facm/1814

Information

Published: March 2020
First available in Project Euclid: 26 October 2019

zbMATH: 07225505
MathSciNet: MR4074393
Digital Object Identifier: 10.7169/facm/1814

Subjects:
Primary: 52C10

Rights: Copyright © 2020 Adam Mickiewicz University

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Vol.62 • No. 1 • March 2020
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