I am interested in the symmetric random walk on $\mathbb{R}$ which increments have the discrete law $$\mu=\sum_{i=1}^q p_i (\delta_{\omega_i}+\delta_{-\omega_i})$$ where the $p_i$ sum to $1/2$ and the $\omega_i$ are real numbers. This problem came up in an unrelated problem, and I was able to obtain the estimates I wanted (basically estimating the probability that the sum is close to 0, see How often a random walk with irrational increments is close to 0?). It basically depends on how well are the ratios between $\omega_i's$ approximable by rational numbers.

I wanted to see if I could attach that to a related body of litterature, but I have not found something relevant. It seems random rotations with irrational increments on the torus have been studied, but mostly for covering problems, and I have not been able to link both questions. I realise there are random walks in general Lie groups, with non-homogeneous increment law, but I am not enough at ease in these fields to see if there could be a connection.

Does anyone think of a (abstract) general problem that has a direct link with my initial problem?