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Let $x$ be an irrational number, and $\beta$ strictly larger than its irrationality index, which means that for some $C>0$, for all $n\in \mathbb{Z}^*$, $$d(nx,\mathbb{Z})>C n^{-\beta}.$$

Are there some ways to obtain bounds on the linear combination of two irrational numbers $x,y$? That is, for $n,m\in \mathbb{Z}^*$, by what could I bound from below $$d(nx+my,\mathbb{Z})?$$ I also assume that $x/y$ is also irrational.

EDIT: I am interested by any (irrational) numbers $x,y$ such that something interesting can be said.

Let $x$ be an irrational number, and $\beta$ strictly larger than its irrationality index, which means that for some $C>0$, for all $n\in \mathbb{Z}^*$, $$d(nx,\mathbb{Z})>C n^{-\beta}.$$ It is known that for a.e. irrational number $x$, the irrationality index is $1$. It is even known that some numbers satisf the above for $\beta=1$ (for instance, $x=\sqrt{2}$).

By arguments from measure theory, I have been able to prove that if $a_n$ satisfies $$\sum_{n=1}^{\infty}na_n<\infty,$$ almost every couple $(x,y)$ of $\mathbb{R}^2$ satisfies for some $C>0$ $$d(nx+my,\mathbb{Z})>Ca_{|n|+|m|},n,m\in \mathbb{Z}.$$ Ideally, I would like, as for a single number $x$, find irrational numbers $(x,y)$ such that this holds in the limiting case $a_n=n^{-2}$.

Has anyone an idea? Or has anyone a useful reference for such things?

Let $x$ be an irrational number, and $\beta$ strictly larger than its irrationality index, which means that for some $C>0$, for all $n\in \mathbb{Z}^*$, $$d(nx,\mathbb{Z})>C n^{-\beta}.$$

Are there some ways to obtain bounds on the linear combination of two irrational numbers $x,y$? That is, for $n,m\in \mathbb{Z}^*$, by what could I bound from below $$d(nx+my,\mathbb{Z})?$$ I also assume that $x/y$ is also irrational.

EDIT: I am interested by any number $x,y$ such that something can be said.

Let $x$ be an irrational number, and $\beta$ strictly larger than its irrationality index, which means that for some $C>0$, for all $n\in \mathbb{Z}^*$, $$d(nx,\mathbb{Z})>C n^{-\beta}.$$

Are there some ways to obtain bounds on the linear combination of two irrational numbers $x,y$? That is, for $n,m\in \mathbb{Z}^*$, by what could I bound from below $$d(nx+my,\mathbb{Z})?$$ I also assume that $x/y$ is also irrational.

EDIT: I am interested by any (irrational) numbers $x,y$ such that something interesting can be said.

Let $x$ be an irrational number, and $\beta$ strictly larger than its irrationality index, which means that for some $C>0$, for all $n\in \mathbb{Z}^*$, $$d(nx,\mathbb{Z})>C n^{-\beta}.$$

Are there some ways to obtain bounds on the linear combination of two irrational numbers $x,y$? That is, for $n,m\in \mathbb{Z}^*$, by what could I bound from below $$d(nx+my,\mathbb{Z})?$$ I also assume that $x/y$ is also irrational.

Let $x$ be an irrational number, and $\beta$ strictly larger than its irrationality index, which means that for some $C>0$, for all $n\in \mathbb{Z}^*$, $$d(nx,\mathbb{Z})>C n^{-\beta}.$$

Are there some ways to obtain bounds on the linear combination of two irrational numbers $x,y$? That is, for $n,m\in \mathbb{Z}^*$, by what could I bound from below $$d(nx+my,\mathbb{Z})?$$ I also assume that $x/y$ is also irrational.

EDIT: I am interested by any number $x,y$ such that something can be said.