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On the distribution of Kloosterman sums. (English) Zbl 1157.11034

Let \(p\) be a prime and \(\mathbb F_p\) denote the field of \(p\) elements. For \(a \in \mathbb F_p^*\), consider the Kloosterman sum \[ K_p(a) = \sum_{x \in \mathbb F_p^*} \exp( 2\pi i(x + ax^{-1})/p), \] where \(x^{-1}\) denotes the multiplicative inverse of \(x\) modulo \(p\). It is known that \(K_p(a)\) is real and satisfies the bound \(| K_p(a)| \leq 2\sqrt p\), so we can define the angles \(\psi_p(a)\) by \[ \psi_p(a) = \arccos ( K_p(a)/2\sqrt p ). \] N. Katz [Gauss sums, Kloosterman sums, and monodromy groups, Princeton Univ. Press (1988; Zbl 0675.14004)] has shown that when \(p\) is a large prime and \(a\) varies in \(\mathbb F_p^*\), these angles are distributed according to the Sato-Tate conjecture. Namely, if \(\mathcal A_p(\alpha, \beta)\) denotes the number of \(a \in \mathbb F_p^*\) with \(\alpha \leq \psi_p(a) \leq \beta\), then \[ \max_{0 \leq \alpha < \beta \leq \pi} \left| \mathcal A_p(\alpha, \beta) - \mu_{ST}(\alpha, \beta)p \right| \ll p^{3/4}, \] where \(\mu_{ST}(\alpha, \beta)\) is the Sato-Tate density \(\mu_{ST}(\alpha, \beta) = \frac 2{\pi} \int_{\alpha}^{\beta} \sin^2 t \, dt\).
In this paper, the author shows that the sums \(x + y\) have a similar distribution when \(x \in \mathcal X\) and \(y \in \mathcal Y\), for sufficiently large sets \(\mathcal X, \mathcal Y \subseteq \mathbb F_p^*\). More precisely, let \(\mathcal W_p(\alpha, \beta; \mathcal X, \mathcal Y)\) denote the number of pairs \((x, y) \in \mathcal X \times \mathcal Y\) such that \(x + y \in \mathcal A_p(\alpha, \beta)\). The author proves that \[ \max_{0 \leq \alpha < \beta \leq \pi} \left| \mathcal W_p(\alpha, \beta; \mathcal X, \mathcal Y) - \mu_{ST}(\alpha, \beta) \#\mathcal X\#\mathcal Y \right| \ll p^{3/4}\sqrt{\#\mathcal X\#\mathcal Y\log p}. \] In particular, one has an asymptotic formula for \(\mathcal W_p(\alpha, \beta; \mathcal X, \mathcal Y)\) provided that \[ \#\mathcal X\#\mathcal Y \gg p^{3/2}(\log p)^{1 + \epsilon} \] for some \(\epsilon > 0\). Furthermore, the author improves on his recent bound [Bull. Aust. Math. Soc. 71, 405–409 (2005; Zbl 1116.11063)] on the nonlinearity of a Boolean function associated with the Kloosterman sums \(K_p(a)\).

MSC:

11L05 Gauss and Kloosterman sums; generalizations
11L40 Estimates on character sums
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
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References:

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