Difference functions of periodic measurable functions

We investigate some problems of the following type: For which sets H is it true that if f is in a given class ℱ of periodic functions and the difference functions ${\displaystyle {\Delta }_{h}f\left(x\right)=f(x+h)-f\left(x\right)}$ are in a given smaller class G for every h ∈ H then f itself must be in G? Denoting the class of counter-example sets by ℌ(ℱ,G), that is, ${\displaystyle ℌ(ℱ,G)=\{H\subset \frac{ℝ}{\mathbb{Z}}:(\exists f\in ℱ G)(\forall h\in H){\Delta }_{h}f\in G\}}$, we try to characterize ℌ(ℱ,G) for some interesting classes of functions ℱ ⊃ G. We study classes of measurable functions on the circle group ${\displaystyle T=\frac{ℝ}{\mathbb{Z}}}$ that are invariant for changes on null-sets (e.g. measurable functions, ${\displaystyle L_p}$, ${\displaystyle L_{\infty }}$, essentially continuous functions, functions with absolute convergent Fourier series (ACF*), essentially Lipschitz functions) and classes of continuous functions on ${\displaystyle T}$ (e.g. continuous functions, continuous functions with absolute convergent Fourier series, Lipschitz functions). The classes ℌ(ℱ,G) are often related to some classes of thin sets in harmonic analysis (e.g. ${\displaystyle ℌ(L_1,\left\{ACF\right\}\cdot )}$ is the class of N-sets). Some results concerning the difference property and the weak difference property of these classes of functions are also obtained.