On the distribution of partial sums of irrational rotations

For an irrational $$\alpha $$α, we investigate the sums $$\sum _{i=1}^n \left( \{i \alpha \} - \frac{1}{2} \right) $$∑i=1n{iα}-12 and $$\sum _{i=1}^n \left\{ \left( \{i \alpha \} - \frac{1}{2} \right) ^2 - \frac{1}{12} \right\} $$∑i=1n{iα}-122-112. We discuss exact formulae and asymptotic estimates for these sums and point out interesting geometrical properties of their graphs in the case when the continued fraction expansion of $$\alpha $$α has a large isolated partial quotient.

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