# A generalisation of Roth's result on Diophantine approximation?

It is a celebrated result of Roth that algebraic numbers cannot be approximated by rationals too accurately: $\newcommand{\norm}[1]{\left\lVert #1 \right\rVert_{\mathbb{R}/\mathbb{T}}}$

Theorem (Roth): Suppose that $\alpha$ is algebraic and irrational. Then for any $\varepsilon > 0$ we have $$\norm{N\alpha} \geq \frac{c_{\alpha,\varepsilon}}{N^{1+\varepsilon}}$$ for some positive constant $c_{\alpha,\varepsilon}$ and for all sufficiently large integers $N$.

Here, $\norm{x}$ denotes the distance of $x$ to $\mathbb{Z}$ (i.e. the distance $x$ from the closest integer).

Is the following statement true?

Conjecture (Generalised Roth): Suppose that $\alpha,\beta$ are algebraic and irrational. Then for any $\varepsilon > 0$ we have $$\norm{N\alpha + \beta} \geq \frac{c_{\alpha,\beta,\varepsilon}}{N^{1+\varepsilon}}$$ for some positive constant $c_{\alpha,\beta,\varepsilon}$ and for all sufficiently large integers $N$.

Some justification:

The conjecture holds if we assume that $\alpha,\beta \in \mathbb{Q}[\sqrt{d}]$, which is quite straightforward (and we can even take $\varepsilon = 0$): Consider $\phi:= N\alpha + \beta - m$ where $m$ is such that $|\phi| = \norm{N\alpha + \beta}$. Multiply $\phi$ by its conjugate (i.e. $\phi$ with each $\sqrt{d}$ replaced by $-\sqrt{d}$) and then approximate the result from below by a positive constant, and from above by $N\norm{N\alpha + \beta}$ times a constant.

Also, for a fixed $\beta$, the conjecture works for almost all $\alpha$ (with respect to Lebesgue measure on $[0,1]$) for the same reasons why the estimate in Roth's theorem works for almost all $\alpha$: Given $N$ and fixing $c$, the estimate fails for a set of $(\alpha,\beta)$ with measure at most $O(\frac{1}{N^{1+\varepsilon}})$. Because $\sum_{N=1}^\infty \frac{1}{N^{1+\varepsilon}} < \infty$, Borel-Cantelli shows that the estimate can only fail for a set of measure $0$.