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On Weyl products and uniform distribution modulo one. (English) Zbl 1410.11106

Sums of the form \(\displaystyle S_N = \sum_{k=1}^N f(x_k),\) where \(f : [0,1] \to {\mathbb R}_0^+\) and \((x_k)_{k \geq 1}\) is a sequence in the unit interval \([0,1)\) are called Weyl sums. These sums play an important role in different scientific branches as the theory of the trigonometric sums, uniformly distributed sequences, number theory and others.
In the present paper, the asymptotic behavior of the trigonometric products of the form \(\displaystyle \prod_{k=1}^N 2 \sin(\pi x_k)\) for \(N \to \infty,\) where the number \(\omega = (x_k)_{k=1}^N\) are evenly distributed in the unit interval \([0,1),\) is studied. The main results are matching lower and upper bounds for such products in the terms of the star-discrepancy of the net \(\omega.\) Two well-known types of uniformly distributed sequences, namely the Kronecker sequence and the van der Corput sequence are used.
In the Introduction of the paper the Weyl product \(\displaystyle P_N = \prod_{k=1}^N f(x_k)\) is considered. A special example of the above product as the lacunary trigonometric product is given. The upper and the lower estimations of this product are discussed. The notion of the star-discrepancy \(D^*_N\) is reminded. Also, here the main results of the paper are presented.
In Theorem 1, a general estimation of the product \(\displaystyle P_N = \prod_{k=1}^N 2 \sin( \pi x_k)\) in the terms of the star-discrepancy \(D^*_N\) of the net \((x_k)_{1 \leq k \leq N}\) is given. A similar result in a weak form have been obtained yet by Hlawka.
The quantity \(\displaystyle P_N^{(d_N)} = \sup_{\omega}\prod_{k=1}^N 2 \sin( \pi x_k)\) is introduced. In Theorem 2, for all \(N\) an upper bound of \(P_N^{(d_N)}\) and for all sufficiently large \(N\) a lower bound of \(P_N^{(d_N)}\) are obtained. The special case when \((x_n)_{n \geq 1}\) is the Kronecker sequence \((\{n\alpha\})_{n \geq 1}\) with irrational number \(\alpha\) is considered.
In Theorem 3, an upper bound of the product \(\displaystyle \prod_{n=1}^{q-1} | 2 \sin (\pi n \alpha)|,\) where \(q\) is the best approximation denominator of \(\alpha,\) is given.
In Theorem 4, the irrational \(\alpha\) is presented by a continued fraction expansion and \(N \in {\mathbb N}\) denotes its Ostrowski expansion. An upper bound of the product \(\displaystyle \prod_{n=1}^N | 2 \sin(\pi n \alpha)|\) in the terms of the best approximation denominator for \(\alpha\) is obtained.
The notion that a real \(\alpha\) is of the type \(t\geq 1\) is reminded. In Corollary 2, \(\alpha\) is of type \(t> 1.\) An upper bound of the product \(\displaystyle \prod_{n=1}^N | 2 \sin(\pi n \alpha)|\) is obtained. Future, the product \(\displaystyle \prod_{n=1}^N | 2 \sin(\pi x_n)|,\) where \((x_n)_{n \geq 1}\) is van der Corput sequence, is studied. In contrast to the Kronecker sequence, the obtained results are very precise.
In Theorem 5, \((x_n)_{n \geq 1}\) is the van der Corput sequence in base 2. It is shown that \[\displaystyle \limsup_{N \to \infty} \frac{1}{N^2} \prod_{n=1}^N | 2 \sin(\pi x_n)| = \frac{1}{2 \pi}.\]
In Theorem 6, \((X_k)_{k \geq 1}\) is a sequence of i. i. d. random variables having uniform distribution on \([0,1). \) The product \(\displaystyle \prod_{k=1}^N | 2 \sin(\pi X_k)\) is considered. The following results are obtained: For all \(\varepsilon > 0\) the upper bound \[ P_n \leq \exp\left(\left(\frac{\pi}{\sqrt{6}} + \varepsilon\right) \sqrt{N \log \log N}\right) \] is almost surely for sufficiently large \(N\), and the lower bound \[ P_n \geq \exp\left(\left(\frac{\pi}{\sqrt{6}} - \varepsilon\right) \sqrt{N \log \log N}\right) \] is trues for infinitely many \(N.\)
In Theorem 7, \(\alpha\) is an irrational number with bounded continued fraction coefficients and \((\xi_n)_{n \geq 1} = (\xi_n(\omega))_{n \geq 1}\) is a sequence of i. i. d. \(\{0,1\}\)-valued random variables with mean \(\displaystyle \frac{1}{2},\) defined on some probability space \((\Omega, {\mathcal A}, {\mathbf P}),\) which indices are a random sequence \((n_k)_{n \geq 1} = (n_k(\omega))_{k \geq 1}\) is the sequence of all numbers \(\{n\geq 1: \xi_n = 1\},\) sorted in increasing order. The product \(\displaystyle P_N = \prod_{k=1}^n 2 \sin(\pi n_k\alpha)\) is considered. It is shown that for all \(\varepsilon > 0\) \({\mathbf P}\)-almost surely the upper bound \[ P_n \leq \exp\left(\left(\frac{\pi}{\sqrt{12}} + \varepsilon\right) \sqrt{N \log \log N}\right) \] is true for all sufficiently large \(N,\) and the lower bound \[ P_n \geq \exp\left(\left(\frac{\pi}{\sqrt{12}} - \varepsilon\right) \sqrt{N \log \log N}\right) \] is true for infinitely many \(N.\)
In Section 2 of the paper, Theorem 1 and 2 are proved. In Section 3, the results for Kronecker sequence are proved. In Section 4, the results on the van der Corput sequence are proved. In Section 5, the probabilistic results are proved.

MSC:

11K06 General theory of distribution modulo \(1\)
11K45 Pseudo-random numbers; Monte Carlo methods
11K36 Well-distributed sequences and other variations
11K60 Diophantine approximation in probabilistic number theory
11J70 Continued fractions and generalizations
11J71 Distribution modulo one
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