The best known example of this is $A=\{ F_0,F_1,\ldots \}$, the Fibonacci numbers, $\alpha=\frac{1+\sqrt{5}}{2}$, the golden ratio. Then $\alpha\ast A \pmod{1}$ has a single limit point (or two, if you think 0 and 1 are different limits modulo 1).

In Representing Ordinal Numbers with Arithmetically Interesting Sets of Real Numbers, Joel Hamkins, Dakota Blair, and I considered this problem under the additional constraint that we considered only the order type of $\alpha\ast A$. In a nutshell, we found that for any countable order types $t_1,\dots,t_k$ and any irrationals $x_1,\dots,x_k$ there is a single set $A$ of positive integers with $x_i\ast A$ having order type $t_i$. There are some other relevant results in that paper. For example, if $A$ has positive density and $\alpha$ is irrational, then $\alpha\ast A$ has an interval of limit points.

Perhaps the most elementary example of an $(\alpha,A)$ pair like this is $A=\{ F_0,F_1,\ldots \}$, the Fibonacci numbers, $\alpha=\frac{1+\sqrt{5}}{2}$, the golden ratio. Then $\alpha\ast A \pmod{1}$ has a single limit point (or two, if you think 0 and 1 are different limits modulo 1).

In Representing Ordinal Numbers with Arithmetically Interesting Sets of Real Numbers, Joel Hamkins, Dakota Blair, and I considered this problem under the additional constraint that we considered only the order type of $\alpha\ast A$. In a nutshell, we found that for any countable order types $t_1,\dots,t_k$ and any irrationals $x_1,\dots,x_k$ there is a single set $A$ of positive integers with $x_i\ast A$ having order type $t_i$. There are some other relevant results in that paper. For example, if $A$ has positive density and $\alpha$ is irrational, then $\alpha\ast A$ has an interval of limit points.