On the Discrepancy of the Sequence (α√n )

Abstract

The first person to consider the discrepancy of sequences of the type (αnσ)n⩾1 (where 0<σ<1) was H. Behnke[1]. The subject was taken up again by one of the authors of this paper[3], who gave a detailed description of the discrepancy's behaviour if either 0<σ<½ or σ=½ and α2∉Q or σ=½ and α−2∈N. In this paper, we study the case of sequences (α√n)n⩾1 where α>0 and α2∈Q. Both limN→∞¯ N‐1/2ω+(α) and limN→∞¯ N‐1/2ω‐(α) are expressed as maxima of finitely many numbers which involve class numbers of imaginary quadratic fields in many cases.

On the discrepancy of the sequence $ \bigl (\alpha \sqrt {n}\bigr) $ II

“…Using Corollary 4 in [1] we get lim sup 70, 1998 On the discrepancy of the sequence À a n p Á II P r o o f. Only the case 4 j p will be proved.…”

“…We shall use the notation of [1] adding D N a sup 0 % x`y % 1 N n1 c xYy À fa n p g Á À Ny À x which was not defined there. We shall use the notation of [1] adding D N a sup 0 % x`y % 1 N n1 c xYy À fa n p g Á À Ny À x which was not defined there.…”