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تسجيل مستخدم جديدA new bound on ErdH{o}s distinct distances problem in the plane over prime fields
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In this paper we obtain a new lower bound on the ErdH{o}s distinct distances problem in the plane over prime fields. More precisely, we show that for any set $Asubset mathbb{F}_p^2$ with $|A|le p^{7/6}$, the number of distinct distances determined by pairs of points in $A$ satisfies $$ |Delta(A)| gg |A|^{frac{1}{2}+frac{149}{4214}}.$$ Our result gives a new lower bound of $|Delta{(A)}|$ in the range $|A|le p^{1+frac{149}{4065}}$. The main tools we employ 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 Stevens and de Zeeuw.
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