Failure of Wiener's property for positive definite periodic functions

We say that Wiener’s property holds for the exponent $p>0$ if we have that whenever a positive definite function $f$ belongs to $L^p(-ϵ,ϵ)$ for some $ϵ>0$, then $f$ necessarily belongs to $L^p(\text{TT})$, too. This holds true for $p\in 2\text{NN}$ 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 $p\mathrm{\notin }2\text{NN}$. Thus we obtain strong extensions of results of Wainger and Shapiro, who proved the negative answer to Wiener’s problem for $p\mathrm{\notin }2\text{NN}$.