Rademacher series with nondifferentiable sums

Abstract

A series of the form Em= amrm(t), where { am I is a sequence of real numbers and rm(t) denotes the mth Rademacher function, sign sin(2mirt), is called a Rademacher series. Although a sufficient condition is known on the order of magnitude of the coefficients {am} in order that f(t) =_ amrm(t) be differentiable almost everywhere, it remains to be determined if this condition is best possible. It is the purpose of this note to give conditions on the order of magnitude of the coefficients of a Rademacher series which are not sufficient in order to have f(t) differentiable almost everywhere. Previously, L. A. Balasov proved [1, p. 631 ] that f(t) has a derivative at at least one point if and only if the limit of { 2mam I exists and is finite. Balasov then showed that this condition on the coefficients is not sufficient for f(t) to be differentiable almost everywhere I', p. 633]. More precisely, he exhibited a sequence {am} for which