On discrepancies of irrational rotations: an approach via rational rotation

Setokuchi and Takashima (Unif Distrib Theory (2) 9:31–57, 2014) and Setokuchi (Acta Math Hung, [11]) gave refinements of estimates for discrepancies by using Schoissengeier’s exact formula. Mori and Takashima (Period Math Hung, [7]) discussed the distribution of the leading digits of \(a^n\) by approximating irrational rotations by “rational rotations”. We apply here their methods to the estimation of discrepancies. We give much more accurate estimates for discrepancies by simple direct calculations, without using Schoissengeier’s formula. We show that the initial segment of the graph of discrepancies of irrational rotations with a single isolated large partial quotient is linearly decreasing, provided we observe the discrepancies on a linear scale with suitable step. We also prove that large hills, caused by single isolated large partial quotients, will appear infinitely often.     

On the distribution of partial sums of irrational rotations

Periodica Mathematica Hungarica - For an irrational $$\alpha $$ , we investigate the sums $$\sum _{i=1}^n \left( \{i \alpha \} - \frac{1}{2} \right) $$ and $$\sum _{i=1}^n \left\{ \left( \{i \alpha...     

Continued fractions and irrational rotations

Let \(\alpha \in (0, 1)\) be an irrational number with continued fraction expansion \(\alpha =[0; a_1, a_2, \ldots ]\) and let \(p_n/q_n= [0; a_1, \ldots , a_n]\) be the nth convergent to \(\alpha \). We prove a formula for \(p_nq_k-q_np_k\) \((k<n)\) in terms of a Fibonacci type sequence \(Q_n\) defined in terms of the \(a_n\) and use it to provide an exact formula for \(\{n\alpha \}\) for all n.