Irregular distribution of {nβ}, n=1,2,3,..., quadrature of singular integrands, and curious basic hypergeometric series

Let βε{lunate}(0,1) be irrational and let {nβ} denote the fractional part of nβ, n≥1. The uniform distribution of {nβ}, n≥1, implies that lim n⇒∞ 1 n ∑ j=1 ng({jβ})= sh{phonetic} 0 1g(t)dt, for each bounded and Riemann integrable g. Hardy and Littlewood proved that this relation persists when g has integrable singularities at 0 and 1, under suitable conditions on g and β. We show that by choosing suitable β, and g with an arbitrarily weak singularity at a suitable interior point gaε{lunate}(0,1), one can ensure that lim n⇒∞ 1 n ∑ j=1 ng({jβ})=∞. On the other hand, if the singularity lies at 0, then at least lim inf n⇒∞ 1 n ∑ j=1 ng({jβ})= sh{phonetic} 0 1g(t)dt. The motivation for these results lies in determination of the radius of convergence of the q or basic hypergeometric series f(z):=1+ ∑ j=1 ∞ { θ k=1 j (A-q^k)}z^j, the solution of the functional equation f(z)(1-zA)=1-zqf(qz). Especially for |A| = |q| = 1, these power series are of interest in Padé approximation. Although the radius of convergence is 1 for "most" A and q, we show that f may be a transcendental entire function for suitable |A| = |q| = 1.