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