A new bound for the Erdős distinct distances problem in the plane over prime fields

Abstract

In this paper we obtain a new lower bound on the Erdős distinct distances problem in the plane over prime fields. More precisely, we show that for any set A ⊂ F 2 p with |A| ≤ p 7/6 and p ≡ 3 mod 4, the number of distinct distances determined by pairs of points in A satisfies |∆(A)| |A| 1 2 + 149 4214 .Our result gives a new lower bound of |∆(A)| in the range |A| ≤ p 1+ 149 4065 . The main tools in our method are the energy of a set on a paraboloid due to Rudnev and Shkredov, a point-line incidence bound given by Stevens and de Zeeuw, and a lower bound on the number of distinct distances between a line and a set in F 2 p . The latter is the new feature that allows us to improve the previous bound due

11. Point-plane incidences and some applications in positive characteristic

“…Remark 13. As this manuscript was being prepared, a better bound Ω(|S| 1128 2107 ) if |S| < p 7 6 for the number of distinct distances defined by a non-isotropic-collinear point set S ⊂ F 2 p was proved by Iosevich et al [21]. The improvement is based on the new observation that if there is a line in the proof of Theorem 12, incident to a large number of points, one can consider distances between points on the line and the rest of S. Dealing with the latter distances enables one to take advantage of Theorem 17, presented in the sequel here.…”

“…The improvement is based on the new observation that if there is a line in the proof of Theorem 12, incident to a large number of points, one can consider distances between points on the line and the rest of S. Dealing with the latter distances enables one to take advantage of Theorem 17, presented in the sequel here. In effect, [21] succeeds in using the incidence bound of Theorem 4 twice, rather than once.…”

“…For more details and references see the recent work of Lewko [30] which specifically uses the energy estimates of [41] presented below for this purpose. Moreover, as it has already been mentioned, energy estimates on P 2 were used by Iosevich et al [21] to get the best known result on the number of distinct distances for sufficiently small sets in F 2 p .…”

New bounds for distance-type problems over prime fields

“…Iosevich, Koh, Pham, Shen, and Vinh [5] improved this result by using a lower bound on the number of distinct distances between a line and a set in F 2 p , and the additive energy of a set on the paraboloid in F 3 p . They proved that for E ⊂ F 2 p with p ≡ 3 mod 4, if |E| ≪ p .…”

Bisectors and Pinned Distances

“…Prior to this paper, the strongest general quantitative bound for small sets was by Stevens and de Zeeuw [16], who proved ∆ pin (A) = Ω(N 1/2+1/30 ) over any field of characteristic not equal to 2. Iosevich, Koh, Pham, Shen, and Vinh [9] recently gave a stronger bound of ∆(A) = Ω(N 1/2+149/4214 ) for point sets in F 2 p for prime p with p ≡ 3 mod 4. Our main result is quantitatively stronger than that of Iosevich et.…”