On linear functional equations modulo $${\mathbb {Z}}$$

Abstract In this paper, we consider the condition $$\sum _{i=0}^{n+1}\varphi _i(r_ix+q_iy)\in {\mathbb {Z}}$$ ∑i=0n+1φi(rix+qiy)∈Z for real valued functions defined on a linear space V . We derive necessary and sufficient conditions satisfying to be decent in following sense: there exist $$f_i:V\rightarrow {R}}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">fi:V→R , $$g_i:V\rightarrow xmlns:mml="http://www.w3.org/1998/Math/MathML">gi:V→Z such that $$\varphi _i=f_i+g_i$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">φi=fi+gi $$(i=0,\dots ,n+1)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">(i=0,⋯,n+1) _{i=0}^{n+1}f_i(r_ix+q_iy)=0$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">∑i=0n+1fi(rix+qiy)=0 all $$x, y\in V$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">x,y∈V