有限域上向量空间大构型的同余类

Pub Date : 2019-01-28 DOI:10.7169/facm/1814

Alex McDonald

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引用次数: 3

摘要

Bennett、Hart、Iosevich、Pakianathan和Rudnev发现了一个指数$s-d$的情况，固定所有的distnace对会导致一个超定系统，因此$q^｛\binom｛k+1｝｛2｝｝$不再是同余类的正确数量。我们确定了正确的数，并证明$|E|\gtrsimq^s$仍然确定了所有同余类的正比例，对于与$k\leqd$情况相同的$s$。

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Congruence classes of large configurations in vector spaces over finite fields

Bennett, Hart, Iosevich, Pakianathan, and Rudnev found an exponent $s d$ 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.

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