Dyadic spike blocks yield counterexamples where lacunary averages diverge to infinity a.e. for mean-zero functions in all L^q and for L^p functions, resolving Erdős problems #995 and #996 negatively.
Rosenthal,On the subspaces of Lp (p > 2)spanned by sequences of independent random variables, Israel J
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Counterexamples for lacunary dilates via dyadic spike blocks
Dyadic spike blocks yield counterexamples where lacunary averages diverge to infinity a.e. for mean-zero functions in all L^q and for L^p functions, resolving Erdős problems #995 and #996 negatively.