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Mat. Zametki, 2015, Volume 98, Issue 4, Pages 606–625 (Mi mz10629)  

This article is cited in 9 scientific papers (total in 9 papers)

Estimates of Trigonometric Sums over Subgroups and Some of Their Applications

Yu. N. Shteinikov

Steklov Mathematical Institute of Russian Academy of Sciences

Abstract: In this paper, we obtain new upper bounds for trigonometric sums over subgroups $\Gamma \subset \mathbb Z_{p}^{*}$ whose size belongs to $[p^{28/95},p^{182/487}]$. Using an approach due to Malykhin, we refine estimates of such sums in $\mathbb Z_{p^{r}}^{*}$ and apply them to the divisibility problem for Fermat quotients.

Keywords: trigonometric sum over a subgroup, Fermat quotient, coset with respect to a subgroup, set with small multiplicative doubling, Abel transformation, Plunnecke's inequality.

Funding Agency Grant Number
Russian Science Foundation 14-11-00433
This work was supported by the Russian Science Foundation under grant 14-11-00433.


DOI: https://doi.org/10.4213/mzm10629

Full text: PDF file (587 kB)
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English version:
Mathematical Notes, 2015, 98:4, 667–684

Bibliographic databases:

UDC: 511.321
Received: 18.11.2014
Revised: 18.03.2015

Citation: Yu. N. Shteinikov, “Estimates of Trigonometric Sums over Subgroups and Some of Their Applications”, Mat. Zametki, 98:4 (2015), 606–625; Math. Notes, 98:4 (2015), 667–684

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. D. P. De Silva, “Estimation of the quantity M-n := min{apn-1(mod p(n))vertical bar 2 <= a <= p - 1}”, Adv. Appl. Math. Sci., 14:6 (2015), 145–154  isi
    2. J. Cilleruelo, M. Z. Garaev, “Congruences involving product of intervals and sets with small multiplicative doubling modulo a prime and applications”, Math. Proc. Camb. Philos. Soc., 160:3 (2016), 477–494  crossref  mathscinet  zmath  isi  elib  scopus
    3. I. V. Vyugin, E. V. Solodkova, I. D. Shkredov, “On the Additive Energy of the Heilbronn Subgroup”, Math. Notes, 101:1 (2017), 58–70  mathnet  crossref  crossref  mathscinet  isi  elib
    4. Yu. N. Shteinikov, “On the product sets of rational numbers”, Proc. Steklov Inst. Math., 296 (2017), 243–250  mathnet  crossref  crossref  mathscinet  isi  elib
    5. T. Cochrane, D. De Silva, Ch. Pinner, “$(p-1)$th roots of unity $\mod p^n$, generalized Heilbronn sums, Lind-Lehmer constants, and Fermat quotients”, Mich. Math. J., 66:1 (2017), 203–219  crossref  mathscinet  zmath  isi
    6. M.-Ch. Chang, B. Kerr, I. E. Shparlinski, “On the exponential large sieve inequality for sparse sequences modulo primes”, J. Math. Anal. Appl., 459:1 (2018), 53–81  crossref  mathscinet  zmath  isi  scopus
    7. S. Macourt, “Bounds on exponential sums with quadrinomials”, J. Number Theory, 193 (2018), 118–127  crossref  mathscinet  zmath  isi  scopus
    8. M. Z. Garaev, “On distribution of elements of subgroups in arithmetic progressions modulo a prime”, Proc. Steklov Inst. Math., 303 (2018), 50–57  mathnet  crossref  crossref  isi  elib
    9. S. Macourt, I. D. Shkredov, I. E. Shparlinski, “Multiplicative energy of shifted subgroups and bounds on exponential sums with trinomials in finite fields”, Can. J. Math.-J. Can. Math., 70:6 (2018), 1319–1338  crossref  mathscinet  isi  scopus
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