Abstract
We study a finite analog of a conjecture of Erd?s on the sum of the squared multiplicities of the distances determined by an n-element point set. Our result is based on an estimate of the number of hinges in spectral graphs.
Highlights
Let q denote the finite field with q elements where q ≫ 1 is an odd prime power
O(V ) and U V are equivalent to the assertion that the inequality |U | ≤ cV holds for some constant c > 0
The notation U = o(V ) is equivalent to the assertion that U = O(V ) but V = O(U ), and the notation U ≪ V is equivalent to the assertion that U = o(V )
Summary
Let q denote the finite field with q elements where q ≫ 1 is an odd prime power. Here, and throughout the paper, the implied constants in the symbols. In [5], Chapman et al took a first step in this direction by showing that if E ⊂ F2q satisfies |E| ≥ q4/3 |∆(E)| ≥ cq This is in line with Wolff’s result for the Falconer conjecture in the plane which says that the Lebesgue measure of the set of distances determined by a subset of the plane of Hausdorff dimension greater than 4/3 is positive. A conjecture of of Erdos [9] on the sum of the squared multiplicities of the distances determined by an n-element point set states that degS(p, r)2 ≤ O n3(log n)α , r>0 p∈S for some α > 0 For this function, Akutsu et al [1] obtained the upper bound. A power of an odd prime, is viewed as an asymptotic parameter
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