Rational approximation using only odd numerators. Dirichlet's simultaneous approximation theorem says:
Given any $n$ real numbers $\alpha_1,\ldots,\alpha_n$ and for every natural number $N \in \mathbb{N}$, there exist integers $q \leq N$, and  $p_1,\ldots,p_n \in \mathbb{Z}$, such that:
$$ \Bigg|\alpha_i - \frac{p_i}{q}\Bigg| < \frac{1}{qN^{1/n}} \text{    for } i=1,\ldots,n $$
I would like to prove a similar theorem, but want to insist that the $p_i$ all be odd.  It's easy to prove a version which has all the $p_i$ even, because the set of points in $\mathbb{R}^n$ with even coordinates is a lattice.  
The general version of Minkowski's Theorem say that a symmetric region containing the origin, if it's volume is sufficiently large, must contain a nonzero lattice point.  But the set of points with odd coefficients is only a coset of the lattice of points with even coefficients.
This seems obviously true that one should be able to replace "lattice" with "coset of the lattice" in Minkowski's Theorem, but I sure don't see how to do it.  I can get the result I want in one dimension by modifying the continued fraction construction, but that doesn't export to $n$ dimensions.  References and insights welcome.
 A: The main point of the Minkowski theorem is that when you take a geometric projection you know that the volume of a certain set is too large not to have the difference of two points be in a given lattice. In symbols if $S$ is any subset of your vector space (more generally locally compact group), $V$, equipped with a Haar measure $m$, and $\Lambda$ is a lattice (more generally co-compact, discrete subgroup) then if $S\subseteq V$ with $m(S)>m(V/\Lambda)$ we have $x\ne y\in S$ so that

$$x\equiv y\mod \Lambda.$$

The Minkowski use of this critical lemma is just that when $S$ is convex and centrally symmetric, then as $2\Lambda$ is also a lattice with

$$m(V/2\Lambda) = 2^nm(V/\Lambda)$$

(in $\Bbb R^n$ with $m$ as Lebesgue measure) when $m(S) > 2^nm(V/\Lambda)$ we can get points $x\equiv y\mod 2\Lambda$. Convexity and symmetry of $S$ gives

$${1\over 2} (x) + {1\over 2} (-y)={1\over 2} (x-y)\in \Lambda\cap S.$$

As a result, your insight about the cosets is exactly on point for this reason because groups are so homogeneous in structure.
But how, exactly, does this work when we're dealing with a non-lattice? In general the proof shows that there is no real control over what coset of a given sub-lattice you get:  it's very general construction using very general assumptions, however since the geometry of the situation is relatively simple, we can exploit it for some progress.
Note that the situation of simultaneous approximation as it manifests in this problem concerns a parallelopiped, namely

$$M^{-1}\bigg(\left[-N-{1\over 2}, N+{1\over 2}\right]\times \left[-N^{-1/d}, N^{-1/d}\right]^{n}\bigg).$$

with

$$M = \begin{pmatrix}
1 && 0 && 0 && \ldots && 0 \\
\alpha_1 && -1 && 0 && \ldots && 0 \\
\alpha_2 && 0 && -1 && \ldots && 0 \\
\vdots && \vdots && \vdots && \ddots && \vdots \\
\alpha_n && 0 && \ldots && 0 && -1
\end{pmatrix}.$$

Luckily here you can easily see how it affects the distance on each coordinate. In particular you can see that if you're willing to have a constant (depending on the dimension only) as a fudge factor, you can expand the result pretty easily just by making your intervals a little wider. Note that the key obstruction to naively doing this is the fact that this is an existence theorem, which means you have essentially no control over what the convergent will look like for any fixed $N$ unless you are willing to expand the hypotheses you're willing to use a little (unless you get lucky and you pick an actual lattice, like when you do all evens). However, if you are happy to accept some of the "on average" machinery of statstical number theory, we note that the density of lattice points is inversely proportional to the determinant of the lattice, hence there must be points in proper cosets--in fact most of them are in cosets! If you want something for a specific choice of coset, eg. all the numerators are odd rather than just some numerator is odd, you can use the fact that you are dealing with a parallelopiped, so that you understand the relative "thickness" in any given direction based on the $\alpha_i$ in question.
