A metric discrepancy result for the sequence of powers of minus two

The law of the iterated logarithm for discrepancies of {(−2)kt}k is proved. This result completes the concrete determination of the law of the iterated logarithm for discrepancies of the geometric progression with integer ratio, and reveals the fact that 2 is the only positive integer θ>1 such that fractional parts of {(−θ)kt}k converge to uniform distribution faster than those of {θkt}k a.e. t.