# Questions tagged [diophantine-approximation]

For questions about approximating real numbers by rational numbers.

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### Rational approximations with denominators restricted to certain residue classes

In certain contexts in Diophantine approximation, it is important for the denominators of the convergents to be restricted to a certain residue class. Consider, for instance, this problem where it is ...
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### Does the set of real numbers with bounded partial quotients have positive measure?

We say a real number $x$ has bounded partial quotients if its continued fraction expansion $[a_0; a_1, a_2 \cdots]$ is bounded by some constant $M=M(x)$. The set $A$ consisting of those numbers whose ...
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### Waring's problem for sum of squares — $\sqrt{2}$ does not lie in a minor arc

Prove that for each sufficiently large integer $N$ there do not exist positive integers $a,q$ with $a<q, (a,q)=1$ such that $q\leq N^{\frac{1}{20}}$ and the distance from $\sqrt{2} - \frac{a}{q}$ ...
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### $\{mb + n \mid b\in \mathbb{R} \setminus \mathbb{Q} \text{ and } m, n \in \mathbb{Z} \}$ is dense in $\mathbb{R}$

I'm fairly new to analysis so this might be very simple. I have found other pages discussing similar questions, but I haven't found any of the answers particularly helpful for me. I've already shown ...
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### How to recover a number from approximation

How can I recover a number $n$ from its approximation $a$. For example, if $n$ is $1.09861228866$, then $a$ can be $1.099$($N$ approximated value). How can I implement a function $F$ that takes $a$, ...
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### Approximating real numbers with rationals up to a constant of the real number.

Say I have a real number $x>0$ and some constant $C>0$. I want to use a rational number $q$ such that the following holds: $|x-q|<Cx$ Of course such numbers $q$ exist. What I want is to ...
### Can every algebraic integer of degree $3$ be approximated by a quotient of linearly recurrent integer sequences of degree $3$?
Given a zero $\alpha \in \mathbb{R}$ of an irreducible monic third degree polynomial $x^3 - a_2x^2 - a_1x - a_0$, are there always integer sequences $(p_n)_{n=1}^\infty$ and $(q_n)_{n=1}^\infty$ ...