A metric discrepancy result for the sequence of powers of minus two (Q2451125)

The author proves a law of the iterated logarithm for discrepancies of the sequence \(\{(-2)^{k}t\}k\geq 1\) modulo 1. This completes earlier results of the author where he considered sequences \(\{\theta^{k}t\}_{k\geq 1}\) modulo 1 (with fixed integer \(\theta > 1\)). In all this work the corresponding \(\limsup\) is explicitly determined. The result shows that \(2\) is the only integer \(\theta > 1\) such that the fractional parts of \(\{(-\theta)^{k}t\}_{k \geq 1}\) converge to uniform distribution faster than those of \(\{\theta^{k}t\}_{k\geq 1}\) (for almost all \(t\)).