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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.

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    $\begingroup$ What do you mean by a "lattice"? Exactly what are you looking for? A function? What should be the domain of the function? $\endgroup$
    – Seva
    Commented Mar 31, 2012 at 9:53
  • $\begingroup$ $F(x) = \sum_{n=-\infty}^\infty f(q^n x)$ ? $\endgroup$
    – Noam D. Elkies
    Commented Mar 31, 2012 at 18:52

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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.

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