For lacunary sequences with a_{n+1} ≥ q a_n, liminf NG_N(x)/log N ≥ 1/2 and limsup ≤ (q+1)/(q-1) a.e.; the limit equals 1 a.e. if a_n divides a_{n+1}.
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Maximal Gaps for Dilated Lacunary Integer Sequences
For lacunary sequences with a_{n+1} ≥ q a_n, liminf NG_N(x)/log N ≥ 1/2 and limsup ≤ (q+1)/(q-1) a.e.; the limit equals 1 a.e. if a_n divides a_{n+1}.