I have the following claim that I think have been proved by someone, but I can not find the reference, hence I would like to ask for help. Here is the claim:

Let $f_1, \ldots, f_n$ be continuous functions from $\mathbb R^d$ to $\mathbb R$ and $L_1, \ldots, L_n$ are $d$-dimensional lattices in $\mathbb R^d$ with the property that $$f_i(x)=f_i(x+l),$$ for all $x$ in $\mathbb R^d$ and all $l$ in $L_i$.

Suppose that $L_i$ and $L_j$ are not in a bigger lattice $L$ for all distinct $i$ and $j$, $f_1, \ldots, f_n$ are linearly independent, or one of them is a constant function.

I am thinking of something along the line of Fourier Transform to solve this problem. Any ideas? Thanks in advance.