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 ∆hf(x) = f(x + h) − f(x) 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, H(F ,G) = {H ⊂ R/Z : (∃f ∈ F \ G) (∀h ∈ H) ∆hf ∈ 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 T = R/Z that are invariant for changes on null-sets (e.g. measurable functions, Lp, L∞, essentially continuous functions, functions with absolute convergent Fourier series (ACF∗), essentially Lipschitz functions) and classes of continuous functions on 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. H(L1,ACF∗) 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.