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Let $\theta_1,\dots,\theta_n$ be algebraic numbers of degree $\leq D$ and Weil height $\leq H$ such that $1,\theta_1,\dots,\theta_n$ are $\mathbb Q$-linearly independent. An effective version of Kronecker's approximation theorem would then assert that, given $\epsilon > 0$ and $\alpha \in \mathbb R^n$, there exist integers $q,p_1,\dots,p_n$ such that $$|q\theta_i - \alpha_i - p_i| \leq \epsilon$$ for all $1 \leq i \leq n$, where $q$ is bounded appropriately in terms of $D$, $H$, and $\epsilon$. I was told that a result like this was originally proved by Erdos and Turan, but I cannot find a reference. Would anyone know of such a reference?

Additionally, I need a following variation of this theorem: let $s,t$ be fixed integers, then given $\epsilon > 0$ and $\alpha \in \mathbb R^n$, there exist integers $q,p_1,\dots,p_n$ such that $$|(sq+t)\theta_i - \alpha_i - p_i| \leq \epsilon$$ for all $1 \leq i \leq n$, where $q$ is bounded appropriately in terms of $D$, $H$, and $\epsilon$. Is something like this known? What are the best known bounds?

I would appreciate any information on this!

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The numbers $\theta_i$ here are real of course. Also there is only one question really; upon replacing $\boldsymbol{\alpha}$ with $(\boldsymbol{\alpha} - t \boldsymbol{\theta})/s$ the second problem reduces to the case that $s = 1$ and $t = 0$.

That being said, there is nothing inherently ineffective in neither of the two best known proofs of Kronecker's approximation theorem: the diophantine approximations one based on the Dirichlet approximation theorem with an inductive scheme on $n$, and the harmonic analysis one based on Weyl's equidistribution criterion. Into either of these proofs you may input Liouville's lower bound on non-vanishing algebraic quantities, in the form $\mathrm{dist}(\mathbf{k} \cdot \boldsymbol{\theta}, \mathbb{Z}) \geq (2n \cdot \| \mathbf{k} \|_{\infty} \cdot H)^{-D^n}$ for non-zero $\mathbf{k} \in \mathbb{Z}^n$ and $\| \mathbf{k} \|_{\infty} := \max_{i=1}^n |k_i|$.

Let me give some indication of how the latter route goes. It is Weyl's theorem that Erdos and Turan made completely explicit; their inequality is where you were referred to. You will however need the extension of the Erdos-Turan estimate to the higher dimensional torus $\mathbb{T}^n = (\mathbb{R}/\mathbb{Z})^n$, which is due to Koksma. A good reference for this is Theorem 1.21 of the book Sequences, Discrepancies and Application by Michael Drmota and Robert Tichy (LNM 1651, 1997). This bounds the discrepancy $D_N := \sup_{I \subset \mathbb{T}^n} \big| \#\{ \mathbf{x}_j \in I, \, j \leq N\} - N \cdot \mathrm{vol}(I) \big|$ of a sequence $\mathbf{x}_j \in \mathbb{T}^n$ by an explicit linear form of its Fourier coefficients:

Theorem. (Erdos, Turan, Koksma). For any sequence $\mathbf{x}_j \in \mathbb{T}^n$ and every $K \in \mathbb{N}$ it holds $$ D_N = D_N(\mathbf{x}) \leq N \cdot (3/2)^n \cdot \Big( \frac{2}{K+1} + \sum_{0 < \|\mathbf{k}\| \leq K} \frac{1}{r(\mathbf{k})} \Big| \frac{1}{N} \sum_{j=1}^N e( \mathbf{k} \cdot \mathbf{x}_j ) \Big| \Big), $$ with $e(z) := e^{2\pi i z}$ and $r(\mathbf{k}) := \prod_{i=1}^n \max (1,|k_i|)$.

Apply this theorem taking $\mathbf{x}_j := j \, \boldsymbol{\theta}$. The exponential sums in this case are geometric series with quotients of the form $e( \mathbf{k} \cdot \boldsymbol{\theta} )$ and $\mathbf{k} \neq \mathbf{0}$. They are bounded in magnitude by $(2 \, \mathrm{dist}(\mathbf{k} \cdot \boldsymbol{\theta}, \mathbb{Z}))^{-1}$, which in turn is bounded by the Liouville diophantine estimate above. This is the same point as in the Polya-Vinogradov inequality on character sums.

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