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We say that Wiener's property holds for the exponent $p>0$ if we have thatwhenever a positive definite function $f$ belongs to $L^p(-\epsilon,\epsilon)$for some $\epsilon>0$, then $f$ necessarily belongs to $L^p(\TT)$, too. Thisholds true for $p\in 2\NN$ by a classical result of Wiener. Recently variousconcentration results were proved for idempotents and positive definitefunctions on measurable sets on the torus. These new results enable us to provea sharp version of the failure of Wiener's property for $p\notin 2\NN$. Thus weobtain strong extensions of results of Wainger and Shapiro, who proved thenegative answer to Wiener's problem for $p\notin 2\NN$.

Failure of Wiener's property for positive definite periodic functions