# invariance under dilations

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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What do you mean by a "lattice"? Exactly what are you looking for? A function? What should be the domain of the function? –  Seva Mar 31 '12 at 9:53
$F(x) = \sum_{n=-\infty}^\infty f(q^n x)$ ? –  Noam D. Elkies Mar 31 '12 at 18:52

## 1 Answer

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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