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.