we have that the function (for suitable f) 


$ F(x)= \sum\_{-\infty}^{\infty}f(x+n) $ is INVARIANT under any integer traslation


$ y=x+n$ for integer 'n'


however my question is can we find a lattice which is invariant under DILATIONS i mean under the transformation $ y=qx$ for integer (positive) or rational 'q' ??


so i am looking a formula like $ F(x)= \sum f(qx) $ so F(x) is invariant under transformation of the form $ y=qx$ thanks.



invariance under dilations
If you lattice $X$ contains at least a point $x$ it must contains all points $qx$ with $q$ rational. Hence $X$ is a dense set. As a consequence if your function $f$ is positive in some interval then function $F$ is infinite everywhere.


