Geodesic
Asymptotics for the sum of powers of distances between power residues modulo a~prime
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Number theory, algebra, and analysis, Tome 276 (2012), pp. 83-95.

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For fixed $q\in(0,4)$, prime $p\to\infty$, and $d\le\exp(c\sqrt{\ln p})$, where $c>0$ is a constant, we obtain the asymptotics for the sum of $q$th powers of distances between neighboring residues of degree $d$ modulo $p$. 
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M. Z. Garaev; S. V. Konyagin; Yu. V. Malykhin. Asymptotics for the sum of powers of distances between power residues modulo a~prime. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Number theory, algebra, and analysis, Tome 276 (2012), pp. 83-95. http://geodesic.mathdoc.fr/item/TM_2012_276_a6/

[1] Arnold V.I., Gruppy Eilera i arifmetika geometricheskikh progressii, MTsNMO, M., 2003

[2] Burgess D.A., “On character sums and primitive roots”, Proc. London Math. Soc. Ser. 3, 12 (1962), 179–192 | DOI | MR | Zbl

[3] Chang M.-C., “On a question of Davenport and Lewis and new character sum bounds in finite fields”, Duke Math. J., 145 (2008), 409–442 | DOI | MR | Zbl

[4] Davenport H., “On the distribution of quadratic residues (mod $p$)”, J. London Math. Soc., 6 (1931), 49–54 | DOI | MR

[5] Khooli K., Primeneniya metodov resheta v teorii chisel, Nauka, M., 1987 | MR

[6] Johnsen J., “On the distribution of powers in finite fields”, J. reine angew. Math., 251 (1971), 10–19 | MR | Zbl

[7] Karatsuba A.A., “Ob odnoi arifmeticheskoi summe”, DAN SSSR, 199:4 (1971), 770–772 | Zbl

[8] Karatsuba A.A., “Summy kharakterov s vesami”, Izv. RAN. Ser. mat., 64:2 (2000), 29–42 | DOI | MR | Zbl

[9] Kurlberg P., Rudnick Z., “The distribution of spacings between quadratic residues”, Duke Math. J., 100:2 (1999), 211–242 | DOI | MR | Zbl