Congruence classes of large configurations in vector spaces over finite\n fields

2019/01/28 by Alex McDonald, McDonald, Alex · 2 citations
Computer Science · Mathematics · #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.1901.09979

openalex publication_date 2019/01/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Bennett, Hart, Iosevich, Pakianathan, and Rudnev found an exponent s<d such\nthat any set E\⊂ mathbbF_q^d with |E| gtrsim q^s determines\n gtrsim q^ binomk+12 congruence classes of (k+1)-point configurations\nfor k\≤ d. Because congruence classes can be identified with tuples of\ndistances between distinct points when k\≤ d, and because there are\n binomk+12 such pairs, this means any such E determines a positive\nproportion of all congruence classes. In the k>d case, fixing all pairs of\ndistnaces leads to an overdetermined system, so q^ binomk+12 is no\nlonger the correct number of congruence classes. We determine the correct\nnumber, and prove that |E| gtrsim q^s still determines a positive proportion\nof all congruence classes, for the same s as in the k\≤ d case.\n

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