A difference property for functions with bounded second differences on amenable topological groups

Let $G$ be a topological group. For a function $f\colon G\to\mathbb R$ and $h\in G$, the right difference function $\Delta_hf$ is defined by $\Delta_hf(g)=f(gh)-f(g)$ ($g\in G$). A function $H\colon G\to\mathbb R$ is said to be additive if it satisfies the Cauchy functional equation $H(g+h)=H(g)+H(h)$ for every $g, h\in G$. A class $F$ of real-valued functions defined on $G$ is said to have the difference property if, for every function $f\colon G\to\mathbb R$ satisfying $\Delta_hf\in F$ for every $h\in G$, there is an additive function $H$ such that $f-H\in F$. The Erdős conjecture claiming that the class of continuous functions on $\mathbb R$ has the difference property was proved by de Bruijn; later on, Carroll and Koehl proved a similar result for the compact Abelian groups and, under an additional assumption, for the compact metric groups, namely, under the assumption that all functions of the form $\nabla_hf(g)=f(hg)-f(g)$, $g\in G$, are Haar measurable for every $h\in G$. One of the consequences of this assumption is the boundedness of the function $\{g,h\}\mapsto f(gh)-f(g)-f(h)+f(e)$, $g,h\in G$, for every function $f$ on a compact group $G$ for which the difference functions $\Delta_hf$ are continuous for every $h\in G$ and the functions $\nabla_hf$ are Haar measurable for every $h\in G$ ($e$ stands for the identity element of the group $G$). In the present paper, we consider the difference property under the very strong assumption that the function $\{g,h\}\mapsto f(gh)-f(g)-f(h)+f(e)$, $g,h\in G$, is bounded. This assumption enables us to obtain results concerning difference properties not only for functions on groups but also for functions on homogeneous spaces.