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

Volume 193 / 2020

Alex Iosevich, Doowon Koh, Thang Pham, Chun-Yen Shen, Le Anh Vinh Acta Arithmetica 193 (2020), 165-174 MSC: Primary 52C10; Secondary 52C35. DOI: 10.4064/aa190214-10-4 Published online: 21 January 2020

Abstract

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 $A\subset \mathbb {F}_p^2$ with $|A|\le p^{7/6}$ and $p\equiv 3\mod 4$, the number of distinct distances determined by pairs of points in $A$ satisfies $$ |\Delta (A)| \gtrsim |A|^{\frac {1}{2}+\frac {149}{4214}}.$$ Our result gives a new lower bound of $|\Delta {(A)}|$ in the range $|A|\le p^{1+\frac {149}{4065}}$.

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 $\mathbb {F}_p^2$. The latter is the new feature that allows us to improve the previous bound due to Stevens and de Zeeuw.

Authors

  • Alex IosevichDepartment of Mathematics
    University of Rochester
    Rochester, NY 14627, U.S.A.
    e-mail
  • Doowon KohDepartment of Mathematics
    Chungbuk National University
    Cheongju, Chungbuk, 28644 Korea
    e-mail
  • Thang PhamDepartment of Mathematics
    University of Rochester
    Rochester, NY 14627 U.S.A.
    e-mail
  • Chun-Yen ShenDepartment of Mathematics
    National Taiwan University
    Taipei, 106 Taiwan
    e-mail
  • Le Anh VinhVietnam Institute of Educational Sciences
    Hanoi, 100000, Vietnam
    e-mail

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