# Questions tagged [diophantine-approximation]

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### Working with truncation of inverse of integers (number of necessary digits)

Ideally I would like to find exact value of $Mr'$ where $r'=\frac{1}{M'}$ where $M'$ ranges from $1$ to $T+1$ and $1\leq M\leq T$ holds. However in real world approximations have finite precision. If ...
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### Chebyshev rational approximation of $e^{x}, x >0$: does it exist?

It's well known that the scalar function $e^x$, for $x \in (-\infty,0]$ can be approximated by Chebyshev rational approximation. In practice, one wants to use a partial fraction decomposition form ...
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### Rational approximation of an integer combination of two irrationals

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 ...
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### Algorithm for comparing $x + y \cdot \log(z)$ for $x,y,z$ rational?

I'm playing with some methods of comparing two real numbers of the form $x + y \log(z)$, where $x,y,z$ are rational numbers and $z$ is positive. There are various estimates on irrationality measures ...
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### $\lfloor \frac{3^n}{2^n} \rfloor = \lfloor \frac{3^n-1}{2^n-1} \rfloor$ for $n\geq2$

I stumbled upon the following claim online: $\lfloor \frac{3^n}{2^n} \rfloor = \lfloor \frac{3^n-1}{2^n-1} \rfloor$ for all integers $n\in \mathbb{N}$, $n\geq2$. Checking with the computer, the claim ...
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### Well distributed sequence uniformly over small intervals

Let $a$ an irrational number. Can we say that there is $c>0$ such that for all integer $k,n$ and $1>u>0$, $$\frac {1}{n}\sum_{i=k}^{k+n} 1_{( (ia) \in [0,u])}< cu ?$$ where (x) is the ...
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### The growth of a sequence related to Liouville numbers [closed]

I am doing a work on Liouville numbers. The Liouville constant $\ell=\sum_{k\geq 0}10^{-k!}$ has its approximation by rational numbers related to the fact that for $v_n=n!$, then $v_{n+1}/v_n$ tends ...
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### Are there infinitely many primes $p$, positive integers $k, n$ such that $1 \le n < p$ and $p^k > n.rad(p^{k+1}−n)$?

Among $168$ prime numbers in range $1$ to $10^3$, there are $84$ prime numbers $n$ such that: $p^k> n.rad(p^{k+1}−n)$ where $1 \le n<p$ and $k=2,3,4$. There are also $84$ prime numbers $n$ such ...
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### Algebraic integers whose matrix representations have singular values in an interval

Let $K$ be a finite extension of $\mathbb{Q}$. Let $\mathcal{O}(K)$ be the ring of integers of $K$. Let $\omega_1,\ldots,\omega_n$ be an integral basis for $K$ over $\mathbb{Q}$. For each $a \in K$...
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### Counterexamples or reasonings about the transcendence of series involving the Möbius function, and polynomials in the denominator

This afternoon I tried to read and understand some sections of the paper Some Applications of Diophantine Approximation by R. Tijdeman procedding from Number Theory for the Millenium III, A K Peters (...
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### The condition on $\alpha$ that $\alpha^n$ is convergent modulo 1

We consider numbers $\alpha\in \mathbb{R}$ with $|\alpha|>1$. Is there any result about a characterization of those $\alpha$ so that $\{\alpha^n\}_{n\in \mathbb{N}}$ is convergent modulo 1? I ...
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### Transcendental functions generating almost integers

Informally speaking, an "almost integer" is a real number very close to an integer. There are some known ways to construct such examples in a systematic way. One is through the use of certain ...
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### Digits in an algebraic irrational number

I am trying to solve a problem and I got a conditional result related to normality of algebraic irrational numbers (Borel conjecture). I know that by using Ridout theorem or Schmidt subspace theorem ...
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### Closest area of research to Transcendental Number Theory or/and Geometry of Numbers?

I am highly interested in doing research in either of 1- Transcendental Number Theory and Algebraic Independence; 2- Diophantine Approximation and Geometry of Numbers. There is no person working ...
### numbers independent over $\mathbb{Q}$ but not BA? numbers that aren't a basis for a number field but are BA?
Has anyone discovered a vector of algebraic real numbers $(a_1,...,a_k)$ such that $1,a_1,...,a_k$ are linearly independent over $\mathbb{Q}$ and such that $(a_1,...,a_k)$ is not "badly approximable"? ...