Limit distributions in metric discrepancy theory (Q1942333)

The authors consider lacunary sequences \((n_k)_{k\geq 1}\) of integers, i.e. sequences satisfying Hadamard's gap condition \(n_{k+1}/n_k\geq q > 1,\quad (k=1,2,\ldots)\). In the main result the limit distribution of \(\sqrt{N}D_N (n_kx)\) (for \(N\to\infty)\) is determined, where \(D_N(n_kx)\) denotes the discrepancy of the sequence \((n_kx)_{k\geq 1}\bmod 1\). The result holds provided that the number of solutions \((k,l)\) of the Diophantine equation \(an_k- bn_l= c\) with \(1\leq k\), \(l \leq N\) is ``small''. The limit distribution is a so-called Kolmogorov distribution with explicitly known densities. The special case \(n_k= a^k\) is studied in more detail and extensions to \(L^2\)-discrepancies are also derived.