New bounds for distance-type problems over prime fields

Distribution of distances in five dimensions and related problems

“…Remark 1.1. It is worth noting that a similar theorem was obtained by Iosevich, Koh and Pham for very small sets in [5], namely, when…”

“…Remark 3.1. We note that one can adapt the methods from [5,6] to prove that the bisector energy is at most the product of |A − A| and the number of collinear triples in A × A, which is bounded by |A − A| • |A| 9/2 . This is slightly weaker than the bound of Lemma 3.2 when |A − A| ∼ |A|.…”

On the Pinned Distances Problem in Positive Characteristic

“…Iosevich, Koh and Pham [18] improved the latter bound in the context of F = F p , p ≡ 3 mod (4) by way of essentially applying the Stevens-de-Zeeuw theorem twice. To bound the number of isosceles triangles they used estimates from [23,38] (also based on the incidence bound from [40]) improving the ∆ pin (A) exponent to 1 2 + 69 1558 = 0.5442 .…”

A Point-Conic Incidence Bound and Applications over $\mathbb F_p$

“…The best current lower bound is |E| 2/3 due to Murphy, Petridis, Pham, Rudnev and Stevens [19]. We also note that it seems very difficult to extend the methods in [12,16,11,19] to the Minkowski and parabola distance functions. We now take advantage of the generality of Theorem 1.1 to study algebraic distances between two sets in F 3 p , where one set lies on a plane and the other set is arbitrary.…”

“…When f (x, y) = x 2 + y 2 , Theorem 6.1 was first proved by Stevens and de Zeeuw in [24] by using a point-line incidence bound. The exponent 8 15 was improved to 1 2 + 149 4214 by Iosevich, Koh, Pham, Shen and Vinh [12], then to 1 2 + 3 74 by Lund and Petridis [16] and to 1 2 + 69 1558 by Iosevich, Koh and Pham [11]. The best current lower bound is |E| 2/3 due to Murphy, Petridis, Pham, Rudnev and Stevens [19].…”