Abstract. We study the Erdös/Falconer distance problem in vector spaces over finite fields. Let F q be a finite field with q elements and takeWe develop a Fourier analytic machinery, analogous to that developed by Mattila in the continuous case, for the study of distance sets in F d q to provide estimates for minimum cardinality of the distance set ∆(E) in terms of the cardinality of E. Bounds for Gauss and Kloosterman sums play an important role in the proof.

If E Ă R 2 is a compact set of Hausdorff dimension greater than 5{4, we prove that there is a point x P E so that the set of distances t|x´y|u yPE has positive Lebesgue measure. 1 arXiv:1808.09346v1 [math.CA] 28 Aug 2018 the complications of replacing the upper Minkowski dimension by the Hausdorff dimension in the claim above.Falconer's distance problem can be thought of as a continuous analogue of a combinatorial problem raised by Erdős in [11]: given a set P of N points in R d , what is the smallest possible cardinality of ∆pP q. A grid is the best known example in all dimensions. In two dimensions, Guth and Katz [15] proved a lower bound for |∆pP q| which nearly matches the grid example (up to a factor of log 1{2 N ). In higher dimensions, there is a larger gap, and the best known result is due to Solymosi and Vu [35]. The Erdős distinct distance problem also makes sense for general norms and much less is known about it. In the planar case, if K is smooth and has strictly positive curvature, the best known bound says that if |P | " N , then |∆ K pP q| Á N 3{4 , with stronger estimates established by Garibaldi in special cases ([13]). There is a conversion mechanism to go from Falconer-type results to Erdőstype results that was developed by the second author together with Hoffman ([17]), Laba ([19]), and Rudnev and Uriarte-Tuero ([18]). It gives estimates for point sets that are fairly spread out. Applying the conversion mechanism to Theorem 1.3 we get the following corollary:Corollary 1.5. Let K be a symmetric convex body in R 2 whose boundary BK is C 8 smooth and has strictly positive curvature. Let P be a set of N points in r0, 1s 2 so that the distance between any two points is Á N´1 {2 . Then there exists x P P such that (1.1) |∆ K,x pP q| Ç N 4 5 .

Abstract. An analog of the Falconer distance problem in vector spaces over finite fields asks for the threshold α > 0 such that |∆(E)| q whenever |E| q α , where E ⊂ F d q , the d-dimensional vector space over a finite field with q elements (not necessarily prime). Here x ∈ E} for a pin y ∈ E has been studied in the Euclidean setting. Peres and Schlag ([25]) showed that if the Hausdorff dimension of a set E is greater than d+1 2 then the Lebesgue measure of ∆y(E) is positive for almost every pin y. In this paper we obtain the analogous result in the finite field setting. In addition, the same result is shown to be true for the pinned dot product set Πy(E) = {x · y : x ∈ E}. Under the additional assumption that the set E has cartesian product structure we improve the pinned threshold for both distances and dot products to

Abstract. We prove a pointwise and average bound for the number of incidences between points and hyperplanes in vector spaces over finite fields. While our estimates are, in general, sharp, we observe an improvement for product sets and sets contained in a sphere. We use these incidence bounds to obtain significant improvements on the arithmetic problem of covering F q , the finite field with q elements, by A · A + · · · + A · A, where A is a subset F q of sufficiently large size. We also use the incidence machinery and develop arithmetic constructions to study the Erdős-Falconer distance conjecture in vector spaces over finite fields. We prove that the natural analog of the Euclidean Erdős-Falconer distance conjecture does not hold in this setting. On the positive side, we obtain good exponents for the Erdős-Falconer distance problem for subsets of the unit sphere in F d q and discuss their sharpness. This results in a reasonably complete description of the Erdős-Falconer distance problem in higher-dimensional vector spaces over general finite fields.

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.