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Let $E \subset {\mathbb F}_q^d$, the $d$-dimensional vector space over the finite field with $q$ elements. Construct a graph, called the distance graph of $E$, by letting the vertices be the elements of $E$ and connect a pair of vertices corresponding to vectors $x,y \in E$ by an edge if $||x-y||:={(x_1-y_1)}^2+\dots+{(x_d-y_d)}^2=1$. We shall prove that the non-overlapping chains of length $k$, with $k$ in an appropriate range, are uniformly distributed in the sense that the number of these chains equals the statistically correct number, $1 \cdot {|E|}^{k+1}q^{-k}$ plus a much smaller remainder.
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참고문헌 (7)
참고문헌 신청
J. Chapman / 2012 / Pinned distance sets, k-simplices, Wolff ’s exponent in finite fields and sum-product estimates / Math Z 271(1-2):63~93
D. Covert / 2010 / Generalized incidence theorems, homogeneous forms and sum-product estimates in finite fields / European J. Combin 31(1):306~319
D. Covert / 2011 / A FurstenbergKatznelson-Weiss type-theorem on (d + 1)-point configurations in sets of positive density in finite field geometries / Discrete Math 311(6):423~430
A. Iosevich / 2007 / Erd˝os distance problem in vector spaces over finite fields / Transactions of the AMS
H. Iwaniec / 2004 / Analytic Number Theory / Colloquium Publications :53
D. Covert / 2010 / Generalized incidence theorems, homogeneous forms and sum-product estimates in finite fields / European J. Combin 31(1):306~319
D. Covert / 2011 / A FurstenbergKatznelson-Weiss type-theorem on (d + 1)-point configurations in sets of positive density in finite field geometries / Discrete Math 311(6):423~430
A. Iosevich / 2007 / Erd˝os distance problem in vector spaces over finite fields / Transactions of the AMS
H. Iwaniec / 2004 / Analytic Number Theory / Colloquium Publications :53
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