The Kronecker-Weyl theorem asserts the following: fix real numbers $\theta\_1,\dots,\theta\_d$, and consider the infinite ray $t(\theta\_1,\dots,\theta\_d)$ $(t\in\Bbb R)$ inside the $d$-dimensional torus $(\Bbb R/\Bbb Z)^d$. Then there exists a subtorus $A$ such that the limiting distribution of $t(\theta\_1,\dots,\theta\_d)$ is uniform on $A$. (In fact, $A$ is the torus defined by any $\Bbb Q$-linear relations among the $\theta\_j$. In other words, the ray is "obviously" confined to this subtorus, and the theorem says that there aren't any other restrictions on where the ray goes.)


I am looking for a reference for this theorem and its proof. It seems to be one of these theorems that is often stated, and called "classical" and "well-known", but almost always without citation. I have been unable to find the theorem proved in this full generality; *a proof under the assumption that the $\theta\_j$ are linearly independent over the rational numbers will not suffice for me*.


I am most interested in a source from the research literature or from a research monograph. But I would also find useful a fully written proof from someone's course notes or the like. Even a published source where this general version was carefully stated (preferably with a definition of "uniformly distributed") might be of some slight use, even if they don't include a proof.



Reference for Kronecker-Weyl theorem in full generality
Why not just prove the result from scratch? It's only a couple of pages and involves only basic Fourier analysis for locally compact abelian groups.


Let
$$\mathbb{T}^n = \left\{(z\_1,\ldots,z\_n) \in \mathbb{C}^n : |z\_l| = 1 \text{ for all $1 \leq l \leq n$}\right\}$$
be the $n$-torus. Let $t\_1, \ldots, t\_n$ be arbitrary real numbers, and let $H$ be the topological closure in $\mathbb{T}^n$ of the subgroup
$$\widetilde{H} = \left\{\left(e^{2\pi i t\_1 y}, \ldots, e^{2\pi i t\_n y}\right) \in \mathbb{T}^n : y \in \mathbb{R}\right\}.$$
The Kronecker--Weyl theorem states that


(1) $H$ is a closed connected subgroup of $\mathbb{T}^n$, namely an $r$-dimensional subtorus of $\mathbb{T}^n$, where $0 \leq r \leq n$ is the dimension over $\mathbb{Q}$ of the span of $t\_1, \ldots, t\_n$, and that


(2) for any continuous function $h : \mathbb{T}^n \to \mathbb{C}$, we have that
$$\lim\_{Y \to \infty} \frac{1}{Y} \int^{Y}\_{0}{h\left(e^{2\pi i t\_1 y}, \ldots, e^{2\pi i t\_n y}\right) \: dy} = \int\_{H}{h(z) \: d\mu\_H(z)},$$
where $\mu\_H$ is the normalised Haar measure on $H$.


To prove this, we begin by observing that $H$ is a closed connected subgroup of $\mathbb{T}^n$ as it is the topological closure of $\widetilde{H}$, which is a subgroup of the compact abelian group $\mathbb{T}^n$, being the image of the continuous group homomorphism $\phi : \mathbb{R} \to \mathbb{T}^n$ given by $\phi(y) = \left(e^{2\pi i t\_1 y}, \ldots, e^{2\pi i t\_n y}\right)$.


Next we recall that a character $\chi : \mathbb{T}^n \to \mathbb{T}$ is of the form
$$\chi(z\_1,\ldots,z\_n) = z\_1^{k\_1} \cdots z\_n^{k\_n}$$
for some $(k\_1,\ldots,k\_n) \in \mathbb{Z}^n$. Conversely, for any $(k\_1,\ldots,k\_n) \in \mathbb{Z}^n$, the function $\chi : \mathbb{T}^n \to \mathbb{T}$ defines a character of $\mathbb{T}^n$. In particular, the dual group of $\mathbb{T}^n$ is isomorphic to $\mathbb{Z}^n$, and hence every character $\chi : \mathbb{Z}^n \to \mathbb{T}$ is of the form
$$\chi(k\_1,\ldots,k\_n) = z\_1^{k\_1} \cdots z\_n^{k\_n}$$
for some $(z\_1,\ldots,z\_n) \in \mathbb{T}^n$.


We claim that the annihilator $H^{\perp}$ of $H$ (namely the set of characters of $\mathbb{T}^n$ that are trivial on $H$) is isomorphic to $\left\{k \in \mathbb{Z}^n : t\_1 k\_1 + \cdots + t\_1 k\_n = 0\right\}$, and consequently $H$ is isomorphic to a torus $\mathbb{T}^r$ for some $0 \leq r \leq n$. Indeed, each character $\chi \in H^{\perp}$ is of the form $\chi(z\_1,\ldots,z\_n) = z\_1^{k\_1} \cdots z\_n^{k\_n}$ for some $(k\_1,\ldots,k\_n) \in \mathbb{Z}^n$ with the property that for all $y \in \mathbb{R}$,
$$1 = \chi\left(e^{2\pi i t\_1 y}, \ldots, e^{2\pi i t\_n y}\right) = e^{2\pi i (t\_1 k\_1 + \cdots + t\_n k\_n) y},$$
and hence $t\_1 k\_1 + \cdots + t\_n k\_n = 0$. Conversely, if $t\_1 k\_1 + \cdots + t\_n k\_n = 0$, then the homomorphism $\chi(z\_1,\ldots,z\_n) = z\_1^{k\_1} \cdots z\_n^{k\_n}$ satisfies $\chi \vert\_{H} = 1$. Now by construction, $H^{\perp}$ is isomorphic to $V \cap \mathbb{Z}^n$ for some vector subspace $V$ of $\mathbb{Q}^n$ of dimension $n - r$, where $r$ is the dimension over $\mathbb{Q}$ of the span of $t\_1, \ldots, t\_n$, and so $H^{\perp} \cong \mathbb{Z}^{n-r}$. Consequently, $\widehat{H} \cong \mathbb{Z}^n / H^{\perp} \cong \mathbb{Z}^r$, as the dual group of $\mathbb{T}^n$ is isomorphic to $\mathbb{Z}^n$, and hence $H \cong \mathbb{T}^r$, thereby proving (1).


For (2), we require the Fourier transform $\widehat{h} : \mathbb{Z}^n \to \mathbb{C}$ of a continuous function $h : \mathbb{T}^n \to \mathbb{C}$, defined by
$$\widehat{h}(\chi) = \int\_{\mathbb{T}^n}{h(z) \overline{\chi(z)} \: dz}.$$
The Poisson summation formula for a closed subgroup of $\mathbb{T}^n$ states that
$$\int\_{H}{h(z) \: d\mu\_H(z)} = \int\_{H^{\perp}}{\widehat{h}(\chi) \: d\mu\_{H^{\perp}}(\chi)},$$
where $\mu\_H$ is the normalised Haar measure on $H$ and $\mu\_{H^{\perp}}$ is the induced Haar measure on $H^{\perp}$, which is the counting measure as $H^{\perp}$ is discrete. Now if $h : \mathbb{T}^n \to \mathbb{C}$ is a trigonometric polynomial, which is to say a function of the form
$$h(z) = \sum\_{k \in \mathbb{Z}^n} c\_k z\_1^{k\_1} \cdots z\_n^{k\_n}$$
for $z = (z\_1, \ldots, z\_n) \in \mathbb{T}^n$, where all but finitely many of the coefficients $c\_k \in \mathbb{C}$ are zero, we claim that
$$\lim\_{Y \to \infty} \frac{1}{Y} \int^{Y}\_{0}{h\left(e^{2\pi i t\_1 y}, \ldots, e^{2\pi i t\_n y}\right) \, dy} = \int\_{H}{h(z) \: d\mu\_H(z)},$$
where $\mu\_H$ is the normalised Haar measure on $H$. From this, we may easily obtain the result in the general case where $h$ is merely a continuous function by the density of the trigonometric polynomials in the space of continuous complex-valued functions on $\mathbb{T}^n$ with regards to the supremum norm. This yields (2).


To prove the claim, we let $\chi : \mathbb{T}^n \to \mathbb{T}$ be a character corresponding to $\widetilde{k} \in \mathbb{Z}^n$. Then
$$\widehat{h}(\chi) = \int\_{\mathbb{T}^n}{h(z) \overline{\chi(z)} \, dz} = \int\_{\mathbb{T}} \hspace{-0.1cm} \cdots \hspace{-0.1cm} \int\_{\mathbb{T}}{\sum\_{k \in \mathbb{Z}^n} c\_k z\_1^{k\_1} \cdots z\_n^{k\_n} \overline{z\_1^{\widetilde{k\_1}} \cdots z\_n^{\widetilde{k\_n}}} \, dz\_1 \cdots dz\_n}.$$
We may interchange the order of summation and integration, as there are only finitely many nonzero members in this sum, and evaluate this integral in order to find that $\widehat{h}(\chi) = c\_{\widetilde{k}}$. Recalling that $H^{\perp}$ is isomorphic to $\left\{k \in \mathbb{Z}^n : t\_1 k\_1 + \cdots + t\_n k\_n \in \mathbb{Z}\right\}$, so that the Haar measure $\mu\_{H^{\perp}}$ on $H^{\perp}$ is simply the counting measure, we therefore obtain by the Poisson summation formula that
$$\int\_{H}{h(z) \: d\mu\_H(z)} = \sum\_{\substack{k \in \mathbb{Z}^n \\ t\_1 k\_1 + \cdots + t\_n k\_n = 0}} c\_k.$$
On the other hand,
\begin{align\*}
\lim\_{Y \to \infty} \frac{1}{Y} \int^{Y}\_{0}{h\left(e^{2\pi i t\_1 y}, \ldots, e^{2\pi i t\_n y}\right) \: dy} & = \lim\_{Y \to \infty} \frac{1}{Y} \sum\_{k \in \mathbb{Z}^n} c\_k \int^{Y}\_{0}{e^{2\pi i (t\_1 k\_1 + \cdots + t\_n k\_n) y} \: dy} \\
& = \sum\_{\substack{k \in \mathbb{Z}^n \\ t\_1 k\_1 + \cdots + t\_n k\_n = 0}} c\_k
\end{align\*}
as required, where we justify the interchanging of order of summation and integration by noting that there being only finitely many nonzero members in this sum.


