Articles of diophantine approximation

Show that $\left| \sqrt2-\frac{h}{k} \right| \geq \frac{1}{4k^2},$ for any $k \in \mathbb{N}$ and $h \in \mathbb{Z}$.

Show that $$\left| \sqrt2-\frac{h}{k} \right| \geq \frac{1}{4k^2},$$ for any $k \in \mathbb{N}$ and $h \in \mathbb{Z}$. I tried many different ways to expand left side and estimate it but always got stuck at some point.

Diophantine approximation – Closest lattice point to a line (2d)

Consider a 2D line $A x + B y + C = 0$ with integer coefficients $A, B, C$. Find the lattice point $(x, y)$ closest to the line, such that $|x|, |y| \leq n$ for some integer $n$. ($x$ and $y$ are integers, of course). It is given that the line intersects the $y$ […]

Estimate number of solutions in the Roth's theorem

There is a fundamental theorem in Diophantine approximation : For algebraic irrational $\alpha$ $$\displaystyle \left \lvert \alpha – \frac{p}{q} \right \rvert < \frac{1}{q^{2 + \epsilon}}$$ with $\epsilon>0$, has finitely many solutions. can we estimate number of solutions $N_{\alpha}(\epsilon)$? for instance : what is the upper bound of $N_{\sqrt{2}}(1)$? number of solutions for $\sqrt{2}$, with $\epsilon=1$. […]

Simultaneous Diophantine approximation: multiple solutions required

Suppose we have $n$ linearly independent (over $\mathbb{Q}$) irrational numbers $\{ \alpha_i | 1\leq i \leq n \}$. For the simultaneous Diophantine approximation problem $$ |q \alpha_i – p_i | < \epsilon , $$ where $q$ and the $p$’s are all integers, we have the LLL algorithm. The problem is, by this algorithm, for each […]

An integral and series to prove that $\log(5)>\frac{8}{5}$

A relationship between $\log(5) \approx 1.6094$ and $\dfrac{3}{2}=1.5$ can be justified by the harmonic approximation $$\log(5) \approx H_2=1+\frac{1}{2}=\frac{3}{2}$$ that can be obtained by regrouping Lehmer’s logarithm $$\log(5) = \sum_{k=0}^\infty \left(\frac{1}{5k+1}+\frac{1}{5k+2}+\frac{1}{5k+3}+\frac{1}{5k+4}-\frac{4}{5k+5}\right)$$ symmetrically around the negative term $$\log(5)=\frac{3}{2}+\sum_{k=1}^\infty \left( \frac{1}{5k-2}+\frac{1}{5k-1}-\frac{4}{5k}+\frac{1}{5k+1}+\frac{1}{5k+2} \right)$$ The corresponding integral is $$\log(5)-\frac{3}{2}=\int_0^1 \frac{x^2(1-x)(1+3x+x^2)}{1+x+x^2+x^3+x^4}\:dx$$ (answer by Olivier Oloa) This is a direct proof […]

Cube roots don't sum up to integer

My question looks quite obvious, but I’m looking for a strict proof for this. (At least, I assume it’s true what I claim.) Why can’t the sum of two cube roots of positive non-perfect cubes be an integer? For example: $\sqrt[3]{100}+\sqrt[3]{4}$ isn’t an integer. Well, I know this looks obvious, but I can’t prove it… […]

Existence of a simultaneous rational approximation of real numbers in (0,1)

I have a simple question the rational approximation of real vectors. Dirichlet’s simultaneous approximation theorem states: Given any $d$ real numbers $\alpha_1,\ldots,\alpha_d$ and for every natural number $N \in \mathbb{N}$, there exists an integer denominator $q \leq N$, and $d$ integer numerators $p_1,\ldots,p_d \in \mathbb{Z}$, such that: $$ \Bigg|\alpha_i – \frac{p_i}{q}\Bigg| < \frac{1}{qN^{1/d}} \text{ for […]

Does every normal number have irrationality measure $2$?

A normal number is a number whose digit expansion in any base is “uniform” in the sense that all finite digit strings occur with the “statistically expected” frequency. I read a sentence somewhere which could be understood as implying that every normal number had irrationality measure $2$. Is it known if all normal numbers have […]

Question on a constructive proof of irrationality of $\sqrt 2$

Here is the constructive proof of $\sqrt 2 \not \in \mathbb Q$ found on this page : Given positive integers $a$ and $b$, because the valuation (i.e., highest power of 2 dividing a number) of $2b^2$ is odd, while the valuation of $a^2$ is even, they must be distinct integers; thus $|2 b^2 – a^2| […]

When is a sequence $(x_n) \subset $ dense in $$?

Weyl’s criterion states that a sequence $(x_n) \subset [0,1]$ is equidistributed if and only if $$\lim_{n \to \infty} \frac{1}{n}\sum_{j = 0}^{n-1}e^{2\pi i \ell x_j} = 0$$ for ever non-zero integer $\ell$. I was wondering if anyone knows of a criterion similar to this one which characterizes when a sequence $(x_n) \subset [0,1]$ is dense in […]