We study the asymptotic behaviour of the sequence of sine products P n (α) = n r=1 |2 sin πrα| for real quadratic irrationals α. In particular, we study the subsequence Q n (α) = qn r=1 |2 sin πrα|, where q n is the nth best approximation denominator of α, and show that this subsequence converges to a periodic sequence whose period equals that of the continued fraction expansion of α. This verifies a conjecture recently posed by Mestel and Verschueren in [15].

We study the asymptotic behavior of Sudler products P N (α) = N r=1 2| sin πrα| for quadratic irrationals α ∈ R. In particular, we verify the convergence of certain perturbed Sudler products along subsequences, and show that lim inf N P N (α) = 0 and lim sup N P N (α)/N = ∞ whenever the maximal digit in the continued fraction expansion of α exceeds 23. This generalizes results obtained for the period one case α = [0; a] in [2].

In this paper we review recently established results on the asymptotic behaviour of the trigonometric product P n (α) = n r=1 |2 sin πrα| as n → ∞. We focus on irrationals α whose continued fraction coefficients are bounded. Our main goal is to illustrate that when discussing the regularity of P n (α), not only the boundedness of the coefficients plays a role; also their size, as well as the structure of the continued fraction expansion of α, is important.

The star discrepancy D * N (P) is a quantitative measure for the irregularity of distribution of a finite point set P in the multi-dimensional unit cube which is intimately related to the integration error of quasi-Monte Carlo algorithms. It is known that for every integer N ≥ 2 there are point sets P in [0, 1) d with |P| = N and D * N (P) = O((log N ) d−1 /N ). However, for small N compared to the dimension d this asymptotically excellent bound is useless (e.g. for N ≤ e d−1 ).In 2001 it has been shown by Heinrich, Novak, Wasilkowski and Woźniakowski that for every integer N ≥ 2 there exist point sets P in [0, 1) d with |P| = N and D * N (P) ≤ C d/N . Although not optimal in an asymptotic sense in N , this upper bound has a much better (and even optimal) dependence on the dimension d.Unfortunately the result by Heinrich et al. and also later variants thereof by other authors are pure existence results and until now no explicit construction of point sets with the above properties is known. Quite recently Löbbe studied lacunary subsequences of Kronecker's (nα)-sequence and showed a metrical discrepancy bound of the form C d(log d)/N with implied absolute constant C > 0 independent of N and d.In this paper we show a corresponding result for digital Kronecker sequences, which are a non-archimedean analog of classical Kronecker sequences.

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