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In the paper by Akos Magyar, Diophantine Equations and Ergodic Theorems, one states in page 923 the following theorem: Theorem 1: Let $Q(m)$ be a nondegenerate polynomial and $\Lambda$ is ...

Consider two numbers $a,b\in R/Z$ and some integer $p\geq 1$. Let $T:L^p(R/Z)\rightarrow L^p(R/Z)$ be the operator given by $$T(f)(x)=1/2(f(x+a)+f(x+b))$$ For which values of $a,b$ do we have almost ...