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 Mat. Sb. (N.S.), 1983, Volume 122(164), Number 4(12), Pages 527–545 (Mi msb2314)  On estimates and asymptotic formulas for rational trigonometric sums that are almost complete

D. A. Mit'kin

Abstract: Suppose that $n\geqslant2$, $q>1$ and $P\geqslant1$ are integers, $P<q$, $f(x)=a_nx^n+…+a_1x$ is a polynomial with integer coefficients, and $(a_n,…,a_2,q)=d$. Hua proved that an incomplete trigonometric sum of the form
$$s(f,q,p)=\sum_{x=1}^pe^{2\pi i\frac{f(x)}q}$$
satisfies the estimate
$$|s(f,q,p)|\ll q^{1-\frac1n+\varepsilon}d^\frac1n\qquad(\varepsilon>0).$$
In this paper sharper estimates are obtained for $n>2$:
$$|s(f,q,p)|\ll q^{1-\frac1n}d^\frac1n$$
and
$$|s(f,q,p)|\ll pq^{-\frac1n+\varepsilon}d^\frac1n+q^{1-\frac1n+\varepsilon}d^\frac1n(\frac qd)^{-\rho},$$
where $\rho=(n-1)/n(n^2-n+1)$. A consequence of the last estimate is that the same type of estimate holds for the number of solutions of the congruence
$$f(x)\equiv c\pmod q;\qquad1\leqslant x\leqslant p.$$
The proofs are based on estimates for complete rational trigonometric sums with prime power denominator which are obtained by Hua's method (this method has also been developed by V. I. Nechaev, C. Chen, S. B. Stechkin and S. V. Konyagin).
Bibliography: 24 titles. Full text: PDF file (780 kB) References: PDF file   HTML file

English version:
Mathematics of the USSR-Sbornik, 1985, 50:2, 513–532 Bibliographic databases:  UDC: 511.3
MSC: Primary 10G10; Secondary 10G05, 10A10

Citation: D. A. Mit'kin, “On estimates and asymptotic formulas for rational trigonometric sums that are almost complete”, Mat. Sb. (N.S.), 122(164):4(12) (1983), 527–545; Math. USSR-Sb., 50:2 (1985), 513–532 Citation in format AMSBIB
\Bibitem{Mit83} \by D.~A.~Mit'kin \paper On estimates and asymptotic formulas for rational trigonometric sums that are almost complete \jour Mat. Sb. (N.S.) \yr 1983 \vol 122(164) \issue 4(12) \pages 527--545 \mathnet{http://mi.mathnet.ru/msb2314} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=725456} \zmath{https://zbmath.org/?q=an:0554.10022|0539.10030} \transl \jour Math. USSR-Sb. \yr 1985 \vol 50 \issue 2 \pages 513--532 \crossref{https://doi.org/10.1070/SM1985v050n02ABEH002842} 

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This publication is cited in the following articles:
1. Shparlinskii I., “Polynomial Congruences”, Acta Arith., 58:2 (1991), 153–156   2. S. V. Konyagin, T. Steger, “On polynomial congruences”, Math. Notes, 55:6 (1994), 596–600     3. Shparlinski I., “On Exponential Sums with Sparse Polynomials and Rational Functions”, J. Number Theory, 60:2 (1996), 233–244    4. Cochrane T., Zheng Z., “A Survey on Pure and Mixed Exponential Sums Modulo Prime Powers”, Number Theory for the Millennium I, eds. Bennett M., Berndt B., Boston N., Diamond H., Hildebrand A., Philipp W., A K Peters, Ltd, 2002, 273–300   •   Contact us: math-net2021_12 [at] mi-ras ru Terms of Use Registration to the website Logotypes © Steklov Mathematical Institute RAS, 2021