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This article is cited in 4 scientific papers (total in 4 papers)
Cycles of Arbitrary Length in Distance Graphs on $\mathbb F_q^d$
A. Iosevich, G. Jardine, B. McDonald Department of Mathematics, University of Rochester, 915 Hylan Building, Rochester, NY, 14627, USA
Abstract:
For $E \subset \mathbb F_q^d$, $d \ge 2$, where $\mathbb F_q$ is the finite field with $q$ elements, we consider the distance graph $\mathcal G^{\text {dist}}_t(E)$, $t\neq 0$, where the vertices are the elements of $E$, and two vertices $x$, $y$ are connected by an edge if $\|x-y\| \equiv (x_1-y_1)^2+\dots +(x_d-y_d)^2=t$. We prove that if $|E| \ge C_k q^{\frac {d+2}{2}}$, then $\mathcal G^{\text {dist}}_t(E)$ contains a statistically correct number of cycles of length $k$. We are also going to consider the dot-product graph $\mathcal G^{\text {prod}}_t(E)$, $t\neq 0$, where the vertices are the elements of $E$, and two vertices $x$, $y$ are connected by an edge if $x\cdot y \equiv x_1y_1+\dots +x_dy_d=t$. We obtain similar results in this case using more sophisticated methods necessitated by the fact that the function $x\cdot y$ is not translation invariant. The exponent $\frac {d+2}{2}$ is improved for sufficiently long cycles.
Received: September 21, 2020 Revised: February 24, 2021 Accepted: June 30, 2021
Citation:
A. Iosevich, G. Jardine, B. McDonald, “Cycles of Arbitrary Length in Distance Graphs on $\mathbb F_q^d$”, Analytic and Combinatorial Number Theory, Collected papers. In commemoration of the 130th birth anniversary of Academician Ivan Matveevich Vinogradov, Trudy Mat. Inst. Steklova, 314, Steklov Math. Inst., Moscow, 2021, 31–48; Proc. Steklov Inst. Math., 314 (2021), 27–43
Linking options:
https://www.mathnet.ru/eng/tm4189https://doi.org/10.4213/tm4189 https://www.mathnet.ru/eng/tm/v314/p31
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Abstract page: | 165 | Full-text PDF : | 33 | References: | 37 | First page: | 7 |
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