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Maybe my answer is beside the point, because the question speaks about the "limit distribution of $t(\theta_1,\dots,\theta_d)$" (for real $t$?). Shouldn't it be $n(\theta_1,\dots,\theta_d)$ for $n=1,2,\ldots$? See https://mathworld.wolfram.com/Kronecker-WeylTheorem.html. The long answer "from scratch" by Peter Humphries proves a different theorem that seems to be more in line with real parameters $t$. Also, the sequence $n(\theta_1,\dots,\theta_d)$ can fill a disconnected manifold, which wouldn't properly be called a "subtorus".

If the question is indeed about the sequence $n(\theta_1,\dots,\theta_d)$, the book by Kuipers and Niederreiter (Uniform Distribution of Sequences, 1974), contains the generic version of what seems to be the Kronecker-Weyl Theorem as Example 6.1 on p. 48. The proper condition is the real numbers $1,\theta_1,\dots,\theta_d$ are linearly independent over the rationals.

The notes on p.51 mention that

a discussion of the exceptional case in this example was also carried out by Weyl,

referring to his classical paper: Hermann Weyl. Über die Gleichverteilung von Zahlen mod. Eins. Mathematische Annalen, 77:313–352, 1916.

Indeed, §5 of that paper, "Die Ausnahmefälle" (the exceptional cases), contains a Theorem 18 (pp. 340-341). It deals with the more general case where each coordinate is not just a linear function of $n$ but an arbitrary polynomial. The conclusion is that the points cover a finite number of affine-linear $r$-dimensional manifolds, (possibly with different integer multiplicities), all these manifolds are parallel, and each of them is filled with uniform density. The theorem specifies how to determine $r$ and the multiplicities.

Here is a statement of this theorem (with different notation), specialized and reformulated for the case of an arithmetic progression $n(\theta_1,\dots,\theta_d)$ as opposed to arbitrary polynomials. The multiplicities are then not necessary.

Let $\vec\theta=(\theta_1,\dots,\theta_d)$. Let $C$ be the set of vectors $\vec x\in\mathbb R^d$ such that $$\langle \vec a,\vec x\rangle\equiv b \pmod 1$$ for all integer vectors $\vec a\in\mathbb Z^d$ and rational numbers $b$ for which the equation $$\langle \vec a,\vec\theta\rangle= b$$ holds. Then all numbers $b$ appearing in these equations have a least common denominator $g$. The sequence $n\vec\theta$ is uniformly distributed modulo 1 in the disjoint union of parallel subspaces $C\cup 2C\cup \dots \cup gC$.

The denominator $g$ is the smallest number $g\ge1$ for which $gC$ contains an integer point (or equivalently, for which $gC$ modulo 1 contains the origin and is therefore equivalent to its corresponding linear subspace $C-C$). If rational dependencies exist only among the numbers $\theta_1,\dots,\theta_d$ and not with the number 1, then the right-hand side $b$ is always $0$, and we set $g=1$.

It is clear that $n\vec\theta$ cycles through the $g$ sets $C,2C,3C,...$. Thus it suffices to look at the generating vector $g\vec\theta$ and prove that its multiples are uniformly distributed modulo one in $gC=C-C$. It is an exercise to reduce this case to the independent case, along the lines of the reduction that I gave for the continuous version. (In fact many of the arguments in that proof appear in Weyl's proof already.)

Maybe my answer is beside the point, because the question speaks about the "limit distribution of $t(\theta_1,\dots,\theta_d)$" (for real $t$?). Shouldn't it be $n(\theta_1,\dots,\theta_d)$ for $n=1,2,\ldots$? See https://mathworld.wolfram.com/Kronecker-WeylTheorem.html. The long answer "from scratch" by Peter Humphries proves a different theorem that seems to be more in line with real parameters $t$. Also, the sequence $n(\theta_1,\dots,\theta_d)$ can fill a disconnected manifold, which wouldn't properly be called a "subtorus".

If the question is indeed about the sequence $n(\theta_1,\dots,\theta_d)$, the book by Kuipers and Niederreiter (Uniform Distribution of Sequences, 1974), contains the generic version of what seems to be the Kronecker-Weyl Theorem as Example 6.1 on p. 48. The proper condition is the real numbers $1,\theta_1,\dots,\theta_d$ are linearly independent over the rationals.

The notes on p.51 mention that

a discussion of the exceptional case in this example was also carried out by Weyl,

referring to his classical paper: Hermann Weyl. Über die Gleichverteilung von Zahlen mod. Eins. Mathematische Annalen, 77:313–352, 1916. DOI:10.1007/BF01475864

Indeed, §5 of that paper, "Die Ausnahmefälle" (the exceptional cases), contains a Theorem 18 (pp. 340-341). It deals with the more general case where each coordinate is not just a linear function of $n$ but an arbitrary polynomial. The conclusion is that the points cover a finite number of affine-linear $r$-dimensional manifolds, (possibly with different integer multiplicities), all these manifolds are parallel, and each of them is filled with uniform density. The theorem specifies how to determine $r$ and the multiplicities.

Here is a statement of this theorem (with different notation), specialized and reformulated for the case of an arithmetic progression $n(\theta_1,\dots,\theta_d)$ as opposed to arbitrary polynomials. The multiplicities are then not necessary.

Let $\vec\theta=(\theta_1,\dots,\theta_d)$. Let $C$ be the set of vectors $\vec x\in\mathbb R^d$ such that $$\langle \vec a,\vec x\rangle\equiv b \pmod 1$$ for all integer vectors $\vec a\in\mathbb Z^d$ and rational numbers $b$ for which the equation $$\langle \vec a,\vec\theta\rangle= b$$ holds. Then all numbers $b$ appearing in these equations have a least common denominator $g$. The sequence $n\vec\theta$ is uniformly distributed modulo 1 in the disjoint union of parallel subspaces $C\cup 2C\cup \dots \cup gC$.

The denominator $g$ is the smallest number $g\ge1$ for which $gC$ contains an integer point (or equivalently, for which $gC$ modulo 1 contains the origin and is therefore equivalent to its corresponding linear subspace $C-C$). If rational dependencies exist only among the numbers $\theta_1,\dots,\theta_d$ and not with the number 1, then the right-hand side $b$ is always $0$, and we set $g=1$.

It is clear that $n\vec\theta$ cycles through the $g$ sets $C,2C,3C,...$. Thus it suffices to look at the generating vector $g\vec\theta$ and prove that its multiples are uniformly distributed modulo one in $gC=C-C$. It is an exercise to reduce this case to the independent case, along the lines of the reduction that I gave for the continuous version. (In fact many of the arguments in that proof appear in Weyl's proof already.)

Maybe my answer is beside the point, because the question speaks about the "limit distribution of $t(\theta_1,\dots,\theta_d)$" (for real $t$?). Shouldn't it be $n(\theta_1,\dots,\theta_d)$ for $n=1,2,\ldots$? See https://mathworld.wolfram.com/Kronecker-WeylTheorem.html. The long answer "from scratch" by Peter Humphries proves a different theorem that seems to be more in line with real parameters $t$. Also, the sequence $n(\theta_1,\dots,\theta_d)$ can fill a disconnected manifold, which wouldn't properly be called a "subtorus".

If the question is indeed about the sequence $n(\theta_1,\dots,\theta_d)$, the book by Kuipers and Niederreiter (Uniform Distribution of Sequences, 1974), contains the generic version of what seems to be the Kronecker-Weyl Theorem as Example 6.1 on p. 48. The proper condition is the real numbers $1,\theta_1,\dots,\theta_d$ are linearly independent over the rationals.

The notes on p.51 mention that

a discussion of the exceptional case in this example was also carried out by Weyl,

referring to his classical paper: Hermann Weyl. Über die Gleichverteilung von Zahlen mod. Eins. Mathematische Annalen, 77:313–352, 1916.

Indeed, §5 of that paper, "Die Ausnahmefälle" (the exceptional cases), contains a Theorem 18 (pp. 340-341). It deals with the more general case where each coordinate is not just a linear function of $n$ but an arbitrary polynomial. The conclusion is that the points cover a finite number of affine-linear $r$-dimensional manifolds, (possibly with different integer multiplicities), all these manifolds are parallel, and each of them is filled with uniform density. The theorem specifies how to determine $r$ and the multiplicities.

Here is a statement of this theorem (with different notation), specialized and reformulated for the case of an arithmetic progression $n(\theta_1,\dots,\theta_d)$ as opposed to arbitrary polynomials. The multiplicities are then not necessary.

Let $\vec\theta=(\theta_1,\dots,\theta_d)$. Let $C$ be the set of vectors $\vec x\in\mathbb R^d$ such that $$\langle \vec a,\vec x\rangle\equiv b \pmod 1$$ for all integer vectors $\vec a\in\mathbb Z^d$ and rational numbers $b$ for which the equation $$\langle \vec a,\vec\theta\rangle= b$$ holds. Then all numbers $b$ appearing in these equations have a least common denominator $g$. The sequence $n\vec\theta$ is uniformly distributed modulo 1 in the disjoint union of parallel subspaces $C\cup 2C\cup \dots \cup gC$.

The denominator $g$ is the smallest number $g\ge1$ for which $gC$ contains an integer point (or equivalently, for which $gC$ modulo 1 contains the origin and is therefore equivalent to its corresponding linear subspace $C-C$). If rational dependencies exist only among the numbers $\theta_1,\dots,\theta_d$ and not with the number 1, then the right-hand side $b$ is always $0$, and we set $g=1$.

It is clear that $n\vec\theta$ cycles through the sets $C,2C,3C,...$. Thus it suffices to look at the generating vector $g\vec\theta$ and prove that its multiples are uniformly distributed modulo one in $gC=C-C$. It is an exercise to reduce this case to the independent case, along the lines of the reduction that I gave for the continuous version. (In fact many of the arguments in that prove appear in Weyl's proof.)

Maybe my answer is beside the point, because the question speaks about the "limit distribution of $t(\theta_1,\dots,\theta_d)$" (for real $t$?). Shouldn't it be $n(\theta_1,\dots,\theta_d)$ for $n=1,2,\ldots$? See https://mathworld.wolfram.com/Kronecker-WeylTheorem.html. The long answer "from scratch" by Peter Humphries proves a different theorem that seems to be more in line with real parameters $t$. Also, the sequence $n(\theta_1,\dots,\theta_d)$ can fill a disconnected manifold, which wouldn't properly be called a "subtorus".

If the question is indeed about the sequence $n(\theta_1,\dots,\theta_d)$, the book by Kuipers and Niederreiter (Uniform Distribution of Sequences, 1974), contains the generic version of what seems to be the Kronecker-Weyl Theorem as Example 6.1 on p. 48. The proper condition is the real numbers $1,\theta_1,\dots,\theta_d$ are linearly independent over the rationals.

The notes on p.51 mention that

a discussion of the exceptional case in this example was also carried out by Weyl,

referring to his classical paper: Hermann Weyl. Über die Gleichverteilung von Zahlen mod. Eins. Mathematische Annalen, 77:313–352, 1916.

Indeed, §5 of that paper, "Die Ausnahmefälle" (the exceptional cases), contains a Theorem 18 (pp. 340-341). It deals with the more general case where each coordinate is not just a linear function of $n$ but an arbitrary polynomial. The conclusion is that the points cover a finite number of affine-linear $r$-dimensional manifolds, (possibly with different integer multiplicities), all these manifolds are parallel, and each of them is filled with uniform density. The theorem specifies how to determine $r$ and the multiplicities.

Here is a statement of this theorem (with different notation), specialized and reformulated for the case of an arithmetic progression $n(\theta_1,\dots,\theta_d)$ as opposed to arbitrary polynomials. The multiplicities are then not necessary.

Let $\vec\theta=(\theta_1,\dots,\theta_d)$. Let $C$ be the set of vectors $\vec x\in\mathbb R^d$ such that $$\langle \vec a,\vec x\rangle\equiv b \pmod 1$$ for all integer vectors $\vec a\in\mathbb Z^d$ and rational numbers $b$ for which the equation $$\langle \vec a,\vec\theta\rangle= b$$ holds. Then all numbers $b$ appearing in these equations have a least common denominator $g$. The sequence $n\vec\theta$ is uniformly distributed modulo 1 in the disjoint union of parallel subspaces $C\cup 2C\cup \dots \cup gC$.

The denominator $g$ is the smallest number $g\ge1$ for which $gC$ contains an integer point (or equivalently, for which $gC$ modulo 1 contains the origin and is therefore equivalent to its corresponding linear subspace $C-C$). If rational dependencies exist only among the numbers $\theta_1,\dots,\theta_d$ and not with the number 1, then the right-hand side $b$ is always $0$, and we set $g=1$.

It is clear that $n\vec\theta$ cycles through the $g$ sets $C,2C,3C,...$. Thus it suffices to look at the generating vector $g\vec\theta$ and prove that its multiples are uniformly distributed modulo one in $gC=C-C$. It is an exercise to reduce this case to the independent case, along the lines of the reduction that I gave for the continuous version. (In fact many of the arguments in that proof appear in Weyl's proof already.)

Maybe my answer is beside the point, because the question speaks about the "limit distribution of $t(\theta_1,\dots,\theta_d)$" (for real $t$?). Shouldn't it be $n(\theta_1,\dots,\theta_d)$ for $n=1,2,\ldots$? See https://mathworld.wolfram.com/Kronecker-WeylTheorem.html. The long answer "from scratch" by Peter Humphries proves a different theorem that seems to be more in line with real parameters $t$. Also, the sequence $n(\theta_1,\dots,\theta_d)$ can fill a disconnected manifold, which wouldn't properly be called a "subtorus".

If the question is indeed about the sequence $n(\theta_1,\dots,\theta_d)$, the book by Kuipers and Niederreiter (Uniform Distribution of Sequences, 1974), contains the generic version of what seems to be the Kronecker-Weyl Theorem as Example 6.1 on p. 48. The proper condition is the real numbers $1,\theta_1,\dots,\theta_d$ are linearly independent over the rationals.

The notes on p.51 mention that

a discussion of the exceptional case in this example was also carried out by Weyl,

referring to his classical paper: Hermann Weyl. Über die Gleichverteilung von Zahlen mod. Eins. Mathematische Annalen, 77:313–352, 1916.

Indeed, §5 of that paper, "Die Ausnahmefälle" (the exceptional cases), contains a Theorem 18 (pp. 340-341). It deals with the more general case where each coordinate is not just a linear function of $n$ but an arbitrary polynomial. The conclusion is that the points cover a finite number of affine-linear $r$-dimensional manifolds, (possibly with different integer multiplicities), all these manifolds are parallel, and each of them is filled with uniform density. The theorem specifies how to determine $r$ and the multiplicities.

Here is a statement of this theorem (with different notation), specialized and reformulated for the case of an arithmetic progression $n(\theta_1,\dots,\theta_d)$ as opposed to arbitrary polynomials. The multiplicities are then not necessary.

Let $\vec\theta=(\theta_1,\dots,\theta_d)$. Let $C$ be the set of vectors $\vec x\in\mathbb R^d$ such that $$\langle \vec a,\vec x\rangle\equiv b \pmod 1$$ for all integer vectors $\vec a\in\mathbb Z^d$ and rational numbers $b$ for which the equation $$\langle \vec a,\vec\theta\rangle= b$$ holds. Then all numbers $b$ appearing in these equations have a least common denominator $g$. The sequence $n\vec\theta$ is uniformly distributed modulo 1 in the disjoint union of parallel subspaces $C\cup 2C\cup \dots \cup gC$.

The denominator $g$ is the smallest number $g\ge1$ for which $gC$ contains an integer point (or equivalently, for which $gC$ modulo 1 contains the origin and is therefore equivalent to its corresponding linear subspace $C-C$). If rational dependencies exist only among the numbers $\theta_1,\dots,\theta_d$ and not with the number 1, then the right-hand side $b$ is always $0$, and we set $g=1$.

It is clear that the sequence $n\vec\theta$ visits the sets $C,2C,3C,\ldots,gC$, not necessarily in this order, and continues cycling through the same sequence of sets over and over. Thus it suffices to look at the generating vector $g\vec\theta$ and prove that its multiples are uniformly distributed modulo one in $gC=C-C$. It is an exercise to reduce this case to the independent case, along the lines of the reduction that I gave for the continuous version. (In fact many of the arguments in that proof appear in Weyl's proof already.)

Maybe my answer is beside the point, because the question speaks about the "limit distribution of $t(\theta_1,\dots,\theta_d)$" (for real $t$?). Shouldn't it be $n(\theta_1,\dots,\theta_d)$ for $n=1,2,\ldots$? See https://mathworld.wolfram.com/Kronecker-WeylTheorem.html. The long answer "from scratch" by Peter Humphries proves a different theorem that seems to be more in line with real parameters $t$. Also, the sequence $n(\theta_1,\dots,\theta_d)$ can fill a disconnected manifold, which wouldn't properly be called a "subtorus".

If the question is indeed about the sequence $n(\theta_1,\dots,\theta_d)$, the book by Kuipers and Niederreiter (Uniform Distribution of Sequences, 1974), contains the generic version of what seems to be the Kronecker-Weyl Theorem as Example 6.1 on p. 48. The proper condition is the real numbers $1,\theta_1,\dots,\theta_d$ are linearly independent over the rationals.

The notes on p.51 mention that

a discussion of the exceptional case in this example was also carried out by Weyl,

referring to his classical paper: Hermann Weyl. Über die Gleichverteilung von Zahlen mod. Eins. Mathematische Annalen, 77:313–352, 1916.

Indeed, §5 of that paper, "Die Ausnahmefälle" (the exceptional cases), contains a Theorem 18 (pp. 340-341). It deals with the more general case where each coordinate is not just a linear function of $n$ but an arbitrary polynomial. The conclusion is that the points cover a finite number of affine-linear $r$-dimensional manifolds, (possibly with different integer multiplicities), all these manifolds are parallel, and each of them is filled with uniform density. The theorem specifies how to determine $r$ and the multiplicities.

Here is a statement of this theorem (with different notation), specialized and reformulated for the case of an arithmetic progression $n(\theta_1,\dots,\theta_d)$ as opposed to arbitrary polynomials. The multiplicities are then not necessary.

Let $\vec\theta=(\theta_1,\dots,\theta_d)$. Let $C$ be the set of vectors $\vec x\in\mathbb R^d$ such that $$\langle \vec a,\vec x\rangle\equiv b \pmod 1$$ for all integer vectors $\vec a\in\mathbb Z^d$ and rational numbers $b$ for which the equation $$\langle \vec a,\vec\theta\rangle= b$$ holds. Then all numbers $b$ appearing in these equations have a least common denominator $g$. The sequence $n\vec\theta$ is uniformly distributed modulo 1 in the disjoint union of parallel subspaces $C\cup 2C\cup \dots \cup gC$.

The denominator $g$ is the smallest number $g\ge1$ for which $gC$ contains an integer point (or equivalently, for which $gC$ modulo 1 contains the origin and is therefore equivalent to its corresponding linear subspace $C-C$). If rational dependencies exist only among the numbers $\theta_1,\dots,\theta_d$ and not with the number 1, then the right-hand side $b$ is always $0$, and we set $g=1$.

It is clear that $n\vec\theta$ cycles through the sets $C,2C,3C,...$. Thus it suffices to look at the generating vector $g\vec\theta$ and prove that its multiples are uniformly distributed modulo one in $gC=C-C$. It is an exercise to reduce this case to the independent case, along the lines of the reduction that I gave for the continuous version. (In fact many of the arguments in that prove appear in Weyl's proof.)