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On the distribution of powers in finite fields. (English) Zbl 0910.11055
Let $$\mathbb F_q$$ be a finite field of characteristic $$p$$ with $$q=p^n$$ elements, $$\{ \omega_1, \ldots , \omega_n \}$$ a basis of $$\mathbb F_q$$ over its prime subfield $$\mathbb F_p$$, and $$\chi$$ a nontrivial multiplicative character of $$\mathbb F_q$$. The author extends the classical result of G. Pólya [Gött. Nachr. 1918, 21–29 (1918; JFM 46.0265.02)] and I. M. Vinogradov [Zh. Fiz.-Mat. O-va Permskiĭ Gos. Univ. 1, 94–98 (1919; JFM 48.1352.04)]: $\left | \sum_{x=1}^{H} \chi(x) \right | \leq p^{1/2} \log p$ for incomplete character sums over $$\mathbb F_p$$ to $\left | \sum_{x \in B} \chi(x) \right | \leq q^{1/2}(1- p/q+ \log q),$ where $$B$$ is a box in $$\mathbb F_q$$ of the form $B= \{x_{1} \omega_{1}+ \cdots + x_{n} \omega_{n} \mid 0 \leq x_{i} < p, \quad 1 \leq i \leq s, \quad 0 \leq x_{s} \leq H < p, \quad x_{s+1}= \cdots =x_{n}=0 \}.$
For the above special box $$B$$, this result gives an improvement of the well-known bound $\left | \sum_{x \in B} \chi(x) \right | \leq q^{1/2}(1+ \log p)^{n}$ obtained by H. Davenport and D. J. Lewis [Rend. Circ. Mat. Palermo (2) 12, 129–136 (1963; Zbl 0119.04302)]. The above result also holds for any translation $$a+B$$ modulo $$p$$ of the box $$B$$.

##### MSC:
 11T24 Other character sums and Gauss sums 11L40 Estimates on character sums 11L26 Sums over arbitrary intervals
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##### References:
  Burde, K., Zur verteilung quadratischer reste, Math. Z., 105, 150-152, (1968) · Zbl 0157.08902  Burde, K., Zur verteilung von potenzresten, J. reine angew. math., 249, 133-172, (1971) · Zbl 0223.10031  Burgess, D.A., On character sums and primitive roots, Proc. London math. soc., 3, 179-192, (1962) · Zbl 0106.04003  Burgess, D.A., A note on the distribution of residues and non-residues, J. London math. soc., 38, 253-256, (1963) · Zbl 0118.04703  Davenport, H.; Lewis, D.J., Character sums and primitive roots in finite fields, Rend. circ. mat. Palermo (2), 12, 129-136, (1963) · Zbl 0119.04302  Perron, O., Bemerkungen über die verteilung der quadratischen reste, Math. Z., 56, 122-130, (1952) · Zbl 0048.03002  Polya, G., Über die verteilung der quadratischen reste und nichtreste, Göttinger nachr., 21-29, (1918) · JFM 46.0265.02  Sokolovskii, A.V., Lower bounds in the large sieve, J. soviet math., 17, 2166-2173, (1981) · Zbl 0467.10036  I. M. Vinogradov, Sur la distribution des residus et des non-residus des puissances, Z. Fiz.-Mat. Obsc. Permsk. Gos. Univ. 1918, 94, 98  A. Winterhof, On the Distribution of Squares in Finite Fields, Bericht, 96/20, Institute für Mathematik, Technische Universität Braunschweig · Zbl 0910.11055
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