On the frequency function of periodic quasianalytic and non-quasianalytic functions

For a real-valued continuous periodic function $f$, let $$A_{r}f(x)=\frac{1}{2r}{\int }_{x-r}^{x+r}f(t)dt,A_{0}f(x)=f(x),$$ and let $r_f(x)$ be the smallest radius at which ${\sup }_{r\ge 0}{A}_{r}f(x)$ is attained. We call $r_f$ the frequency function of $f$. Steinerberger proved that a two-valued condition involving both $r_f$ and $r_{-f}$ characterizes a single sinusoidal function, and asked if the same result holds under the assumption that $r_f$ has finite range (or even range with Lebesgue measure zero). In this paper, we prove that a stronger finite-valued condition for $r_f$ characterizes a sinusoidal function for the case when $f$ is periodic, smooth, and belongs to a quasianalytic Denjoy-Carleman class. However, we also show that the conjecture of Steinerberger is not true in general, since we can construct non-sinusoidal periodic smooth functions in non-quasianalytic Denjoy-Carleman classes such that their frequency functions have finite range.