Group actions and geometric combinatorics in 𝔽qd$\mathbb{F}_{q}^{d}$

Abstract

General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Abstract. In this paper we apply a group action approach to the study of Erdős-Falconer type problems in vector spaces over finite fields and use it to obtain non-trivial exponents for the distribution of simplices. We prove that there exists, where T d k (E) denotes the set of congruence classes of k-dimensional simplices determined by k + 1-tuples of points from E. Non-trivial exponents were previously obtained by Chapman, Erdogan, Hart, Iosevich and Koh2 (E) in the plane was obtained by Bennett, Iosevich and Pakianathan ([1]). These results are significantly generalized and improved in this paper. In particular, we establish the Wolff exponent , previously established in [4] for the q ≡ 3 mod 4 case to the case q ≡ 1 mod 4, and this results in a new sum-product type inequality. We also obtain non-trivial results for subsets of the sphere in F d q , where previous methods have yielded nothing. The key to our approach is a group action perspective which quickly leads to natural and effective formulae in the style of the classical Mattila integral from geometric measure theory.

Congruence classes of large configurations in vector spaces over finite fields

“…In [2], Chapman, Erdogan, Hart, Iosevich, and Koh study the d = 2 case, and prove that if q ≡ 3 mod 4 one has |∆(E)| q whenever |E| q 4/3 . The same exponent was later obtained by Bennett, Hart, Iosevich, Pakianathan, and Rudnev in [1] without the assumption that q ≡ 3 mod 4. Several interesting variants of the distance problem have also been studied in the finite field context.…”

“…However, by inspection, we see this exponent is only non-trivial if d > k+1 2 . Bennett, Hart, Iosevich, Pakianathan, and Rudnev later found in [1] that one can recover a positive proportion of all congruence classes, for any d ≥ k ≥ 2, if |E| q d− d−1 k+1 . We observe that this exponent is always non-trivial.…”

Areas of Triangles and SL2 Actions in Finite Rings

“…However, in order for this result to be non-trivial the exponent must be < d, and that only happens when k+1 2 < d. So, the result is limited to fairly small configurations. This result is improved in [2] by Bennett, Hart, Iosevich, Pakianathan, and Rudnev, who prove that for any k ≤ d a set E ⊂ F d q determines a positive proportion of all congruence classes of (k + 1)-point configurations provided |E| q d− d−1 k+1 . This exponent is clearly non-trivial for all k. In [11], I extended this result to the case k ≥ d.…”

The analog of the Erdös distance problem in finite fields

“…We would like to mention that in a forthcoming paper [1], Bennett, Hart, Iosevich, Pakianathan and Rudnev have employed some elementary arguments while establishing the lower bound |T d…”