I apologise, no one seems interested in this question. But for completeness sake, in answer to the first question, it seems that for Riemann Theta functions we define $n \cdot \Omega \cdot n = \Sigma_{i, j = 1}^{g} \Omega_{ij}n_{i}n_{j}$ giving a scalar, where $n \in \mathbb{Z}^{g}$ and $\Omega \in \mathbb{C}^{g \times g}$. I assume $(\xi + m_{1})^{T}(z + m_{2})$ is just the dot product also giving a scalar.

I found the original paper of Umemura online so have edited the question to take this into account. I am still struggling to understand, but I think everything is there, if someone would be kind enough to explain.

Sorry if I am flogging a dead horse, but what happens if we require the limit to be equal for all irrational $x$? i.e. if $lim_{n \rightarrow \infty}$$\frac{1}{N}$$card${$k : 1 \leq k \leq N, \tilde{(kx)} \in A$} = $lim_{n \rightarrow \infty}$$\frac{1}{N}$$card${$k : 1 \leq k \leq N, \tilde{(ky)} \in A$} for all $x, y$ irrational, then does the limit equal $\mu(A)$?

A bit more general: Vitali Set: one element chosen from each coset of $\mathbb{R}$ modulo a countable dense subgroup of $\mathbb{R}$, Bernstein Set: a set $E \subset \mathbb{R}$ such that both $E$ and its complement are totally imperfect in $\mathbb{R}$.

Thanks Stefan for asking this. I am curious about the same question with regards to Bernstein sets instead of Vitali sets. Further, can one have Vitali sets without Bernstein? Don't know whether to ask this as a separate question...

@Joel... I wasn't registered so couldn't vote... I am now...