Geodesic
Cycles of Arbitrary Length in Distance Graphs on $\mathbb F_q^d$
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic and Combinatorial Number Theory, Tome 314 (2021), pp. 31-48.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{TM_2021_314_a1,
     author = {A. Iosevich and G. Jardine and B. McDonald},
     title = {Cycles of {Arbitrary} {Length} in {Distance} {Graphs} on $\mathbb F_q^d$},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {31--48},
     publisher = {mathdoc},
     volume = {314},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2021_314_a1/}
}
TY  - JOUR
AU  - A. Iosevich
AU  - G. Jardine
AU  - B. McDonald
TI  - Cycles of Arbitrary Length in Distance Graphs on $\mathbb F_q^d$
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2021
SP  - 31
EP  - 48
VL  - 314
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2021_314_a1/
LA  - ru
ID  - TM_2021_314_a1
ER  - 
%0 Journal Article
%A A. Iosevich
%A G. Jardine
%A B. McDonald
%T Cycles of Arbitrary Length in Distance Graphs on $\mathbb F_q^d$
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2021
%P 31-48
%V 314
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2021_314_a1/
%G ru
%F TM_2021_314_a1
A. Iosevich; G. Jardine; B. McDonald. Cycles of Arbitrary Length in Distance Graphs on $\mathbb F_q^d$. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic and Combinatorial Number Theory, Tome 314 (2021), pp. 31-48. http://geodesic.mathdoc.fr/item/TM_2021_314_a1/

[1] Bennett M., Chapman J., Covert D., Hart D., Iosevich A., Pakianathan J., “Long paths in the distance graph over large subsets of vector spaces over finite fields”, J. Korean Math. Soc., 53:1 (2016), 115–126 | DOI | MR | Zbl

[2] Bennett M., Hart D., Iosevich A., Pakianathan J., Rudnev M., “Group actions and geometric combinatorics in $\mathbb {F}_q^d$”, Forum math., 29:1 (2017), 91–110 | DOI | MR | Zbl

[3] Bennett M., Iosevich A., Pakianathan J., “Three-point configurations determined by subsets of $\mathbb {F}_q^2$ via the Elekes–Sharir paradigm”, Combinatorica, 34:6 (2014), 689–706 | DOI | MR | Zbl

[4] Chapman J., Erdoğan M.B., Hart D., Iosevich A., Koh D., “Pinned distance sets, $k$-simplices, Wolff's exponent in finite fields and sum–product estimates”, Math. Z., 271:1–2 (2012), 63–93 | DOI | MR | Zbl

[5] Covert D., Hart D., Iosevich A., Koh D., Rudnev M., “Generalized incidence theorems, homogeneous forms and sum–product estimates in finite fields”, Eur. J. Comb., 31:1 (2010), 306–319 | DOI | MR | Zbl

[6] Greenleaf A., Iosevich A., Taylor K., “Configuration sets with nonempty interior”, J. Geom. Anal., 31:7 (2021), 6662–6680 ; arXiv: 1907.12513 [math.CA] | DOI | MR | Zbl

[7] Greenleaf A., Iosevich A., Taylor K., On $k$-point configuration sets with nonempty interior, E-print, 2020, arXiv: 2005.10796 [math.CA] | MR | Zbl

[8] Hart D., Iosevich A., “Sums and products in finite fields: An integral geometric viewpoint”, Radon transforms, geometry, and wavelets, Contemp. Math., 464, Amer. Math. Soc., Providence, RI, 2008, 129–135 | DOI | MR | Zbl

[9] Hart D., Iosevich A., Koh D., Rudnev M., “Averages over hyperplanes, sum–product theory in vector spaces over finite fields and the Erdős–Falconer distance conjecture”, Trans. Amer. Math. Soc., 363:6 (2011), 3255–3275 | DOI | MR | Zbl

[10] Iosevich A., Parshall H., “Embedding distance graphs in finite field vector spaces”, J. Korean Math. Soc., 56:6 (2019), 1515–1528 | MR | Zbl

[11] Iosevich A., Rudnev M., “Erdős distance problem in vector spaces over finite fields”, Trans. Amer. Math. Soc., 359:12 (2007), 6127–6142 | DOI | MR | Zbl

[12] Iosevich A., Taylor K., Uriarte-Tuero I., “Pinned geometric configurations in Euclidean space and Riemannian manifolds”, Mathematics, 9:15 (2021), 1802 ; arXiv: 1610.00349 [math.CA] | DOI

[13] Minkowski H., “Grundlagen für eine Theorie der quadratischen Formen mit ganzzahligen Koeffizienten”, Gesammelte Abhandlungen, B. G. Teubner, Leipzig, 1911, 3–145 | MR

[14] Pham D.H., Pham T., Vinh L.A., “An improvement on the number of simplices in $\mathbb {F}_q^d$”, Discrete Appl. Math., 221 (2017), 95–105 | DOI | MR | Zbl

[15] Phong D.H., Stein E.M., “Radon transforms and torsion”, Int. Math. Res. Not., 1991:4 (1991), 49–60 | DOI | MR | Zbl

[16] Shteinikov Yu.N., “Bolshie puti v distantsionnykh grafakh v vektornykh prostranstvakh nad konechnym polem”, Chebyshev. sb., 19:3 (2018), 311–317 | Zbl

[17] Sogge C.D., Fourier integrals in classical analysis, Cambridge Tracts Math., 105, Cambridge Univ. Press, Cambridge, 1993 | MR | Zbl

[18] Vinh L.A., “On a Furstenberg–Katznelson–Weiss type theorem over finite fields”, Ann. Comb., 15:3 (2011), 541–547 | DOI | MR | Zbl