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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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