I have a continuous function $f$ on a locally compact Abelian group $G$ with compact support, and I would like to say that the zeroes of $f$ are sparse in some sense (isolated would be good, uniformly discrete would be great).

~~ Now, if $G=\mathbb R^d$, then this is a consequence of the Paley-Wiener theorem, but $G$ is a general LCAG.~~

~~ Now, for what I am doing I can replace $G$ with the subgroup generated by $\sup(f)$, thus by the structure theorem I can assume that $G$ has the form $\mathbb R^d \times \mathbb Z^n \times K$. Also, by a simple trick I am sure I can ignore the $\mathbb Z^n$ component. If $K$ was not there, I would be done, but I don't see any way of eliminating it. ~~

~~ Anyhow, since the dual of $K$ is discrete, intuitively anything in here is isolated and Paley-Wiener should solve the problem in $\mathbb R^n$. Unfortunately, it looks like this intuitive part becomes a proof only for functions $f: \mathbb R^d \times K \to \mathbb C$ of the form
$$
f(s,t)=g(s)h(t)
$$~~

So my questions are:

1) Is there any general result of the type I am seeking in the case of LCAG? I know few "uncertainty principle" type results, which unfortunately are not what I need but maybe there is a variant I am not familiar with which would work.

~~2) Is there any Paley -Wiener Theorem for the case $G= \mathbb R^d \times K$, where $K$ is any compact group? [ Note that $K$ need not be Lie].~~

**Edit:** As the comments already provide a counterexample, is the following weaker version true, at least in $\mathbb R^d$?

Question: Let $f$ be a continuous function with compact support, which is positive definite. Can we show that there exists $t_1,..,t_k$ such that $\sum T_{t_i} (\widehat{f})$ is nowhere vanishing, where $T_{t_i}$ denotes translation by $t_i$?