Ultra-short sums of trace functions. (English) Zbl 1530.11096
In 1926, the Dutch mathematician Hendrik Kloosterman, introduced the exponential type of sums over finite fields later known as Kloosterman sums. The other famous sums are Gauss and Jacobian. All sums are based on trace functions of primitive roots of unity over a finite field. Kloosterman sums have many applications in analytic number theory and coding theory.
The normalized Kloosterman sums modulo a prime number q is given by \[Kl_{2}(a; q) = \dfrac{1}{\sqrt{q}}\sum_{x\in\mathbb{F}_{q}^{\times}}e(\dfrac{ax+\bar{x}}{q}), \quad e(z) = e^{2i\pi z},\] where \(\bar{x}\) is the multiplicative inverse of \(x\).
The main theorem in the article is as follows:
Theorem (Ultra-short sums of additive characters and Kloosterman sums).
Let \(g \in \mathbb{Z}[X]\) be a fixed monic polynomial of degree \(d \geq 1\). For any field \(K\), denote by \(Z_{g}(K)\) the set of zeros of \(g\) in \(K\), and put \(Zg = Zg(\mathbb{C})\). Let \(K_{g} = \mathbb{Q}(Z_{g})\) be the splitting field of \(g\).
(1) As \(q \longrightarrow \infty\) through prime numbers unramified and totally split in \(K_{g}\), the sums \[ \sum_{x \in \mathbb{Z}_{g}(\mathbb{F}_{q})}e\left(\dfrac{ax}{q}\right) \tag{\(\ast\)} \] parameterized by \(a \in \mathbb{F}_{q}\) become equidistributed in \(\mathbb{C}\) with respect to some explicit probability measure \(\mu_{g}\).
(2) Suppose that \(0 \notin \mathbb{Z}_{g}\). As \(q \longrightarrow \infty\) through prime numbers unramified and totally split in \(K_{g}\), the sums \[ \sum_{x \in \mathbb{Z}_{g}(\mathbb{F}_{q})}Kl_{2}(ax; q) \] parameterized by \(a \in \mathbb{F}_{q}\) become equidistributed in \(C\) with respect to the measure which is the law of the sum of \(d\) independent Sato-Tate random variables.
In this article, the authors have incorporated the topics from modern algebra, analytic number theory, topology and mathematical statistics. Also, they have used the open-source software sagemath to demonstrate the distributions of the sums(*) for different polynomials \(g\) and different primes \(q\) with respect to the probability measure \(\mu_{g}\). This article has also provided some more propositions and examples. I should appreciate the authors for the efforts they have taken to bring out such a great paper.
The normalized Kloosterman sums modulo a prime number q is given by \[Kl_{2}(a; q) = \dfrac{1}{\sqrt{q}}\sum_{x\in\mathbb{F}_{q}^{\times}}e(\dfrac{ax+\bar{x}}{q}), \quad e(z) = e^{2i\pi z},\] where \(\bar{x}\) is the multiplicative inverse of \(x\).
The main theorem in the article is as follows:
Theorem (Ultra-short sums of additive characters and Kloosterman sums).
Let \(g \in \mathbb{Z}[X]\) be a fixed monic polynomial of degree \(d \geq 1\). For any field \(K\), denote by \(Z_{g}(K)\) the set of zeros of \(g\) in \(K\), and put \(Zg = Zg(\mathbb{C})\). Let \(K_{g} = \mathbb{Q}(Z_{g})\) be the splitting field of \(g\).
(1) As \(q \longrightarrow \infty\) through prime numbers unramified and totally split in \(K_{g}\), the sums \[ \sum_{x \in \mathbb{Z}_{g}(\mathbb{F}_{q})}e\left(\dfrac{ax}{q}\right) \tag{\(\ast\)} \] parameterized by \(a \in \mathbb{F}_{q}\) become equidistributed in \(\mathbb{C}\) with respect to some explicit probability measure \(\mu_{g}\).
(2) Suppose that \(0 \notin \mathbb{Z}_{g}\). As \(q \longrightarrow \infty\) through prime numbers unramified and totally split in \(K_{g}\), the sums \[ \sum_{x \in \mathbb{Z}_{g}(\mathbb{F}_{q})}Kl_{2}(ax; q) \] parameterized by \(a \in \mathbb{F}_{q}\) become equidistributed in \(C\) with respect to the measure which is the law of the sum of \(d\) independent Sato-Tate random variables.
In this article, the authors have incorporated the topics from modern algebra, analytic number theory, topology and mathematical statistics. Also, they have used the open-source software sagemath to demonstrate the distributions of the sums(*) for different polynomials \(g\) and different primes \(q\) with respect to the probability measure \(\mu_{g}\). This article has also provided some more propositions and examples. I should appreciate the authors for the efforts they have taken to bring out such a great paper.
Reviewer: Telveenus Antony (Kingston upon Thames)
Keywords:
equidistribution; linear relations between algebraic numbers; Weyl sums; roots of polynomial congruences; trace functionsSoftware:
SageMathReferences:
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