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8
votes
2answers
499 views

Does the Galois group of a Pisot polynomial contain the alternating group?

Let $n \in \mathbb{N}$, and let $p(X) \in \mathbb{Z}[X]$ be a monic polynomial of degree $n$. Suppose that exactly one complex root of $p$ is of modulus $> 1$, and that the remaining $n-1$ roots of ...
4
votes
0answers
80 views

Can a product of conjugates be a Pisot number again?

Let $p(X) \in \mathbb{Z}[X]$ be an irreducible polynomial, and let $\alpha_1 \dots, \alpha_n$ be its roots in $\mathbb{C}$. Suppose that $\alpha_1$ is a Pisot number (that is, $\alpha_1 \in ...
1
vote
1answer
151 views

Central binomials and irrationals

I suppose this is une cause perdue, but it would be nice if the following held. Let $\theta$ be an irrational, and let $c_m = {{2m}\choose m}$, the central binomial. For a real number, and let $d(r)$ ...
8
votes
3answers
288 views

“Most Similar Vector Problem” on an Integer Lattice?

I am currently working on problem that I think could be expressed as an integer lattice problem. Given $u \in \mathbb{R}^n$ and a bounded integer lattice $L = \mathbb{Z}^n \cap [-M,M]^n$ I would like ...
2
votes
0answers
122 views

binomial coefficients and irrationals

The following, probably either currently impossible to deal with, or having a negative solution, arose from an ergodic theory question, presumably itself currently intractible. I am not a number ...
15
votes
1answer
883 views

Why is there a $\sqrt{5}$ in Hurwitz's Theorem?

Hurwitz's theorem is an extension of Minkowski's Theorem and deals with rational approximations to irrational numbers. The theorem states: For every irrational number $\alpha$, there are infinitely ...
9
votes
1answer
458 views

A weakening of the Littlewood conjecture

For real numbers $x$, let $\|x\|$ denote the distance from $x$ to the nearest integer. Define a function $\ell:\mathbb{R}^2\rightarrow\mathbb{R}$ by ...
10
votes
0answers
85 views

Looking for an effective irrationality measure of $\pi$

Most standard summaries of the literature on irrationality measure simply say, e.g., that $$ \left| \pi - \frac{p}{q}\right| > \frac{1}{q^{7.6063}} $$ for all sufficiently large $q$, without giving ...
0
votes
0answers
90 views

Rate of convergence of an algebraic irrational rotation

Let $\alpha \in \mathbb{S}^1$ be an algebraic number with $\mathop{\mathrm{arg}}(\alpha)/\pi$ irrational. Is it possible for the rotation by $\alpha$ to converge exponentially fast to a $\xi \in ...
3
votes
1answer
110 views

Positive rational numbers as sum of unit fractions [duplicate]

Let $U = \{\frac{1}{n}: n\in\mathbb{N}, n>0\}$ be the set of unit fractions. For integers $m,n>0$ there is always a finite subset $S\subseteq U$ such that $\frac{m}{n} = \sum_{u\in S} u$, see ...
3
votes
1answer
129 views

Simultaneous approximation by rationals with relatively prime numerators

The following seems hard to me (or perhaps just not true), but perhaps I am mistaken. It is known that given irrational numbers $x_1$ and $x_2$, there are infinitely many simultaneous rational ...
1
vote
0answers
52 views

Big Omega result about number of totally positive integers with fixed trace

There is much literature on the study of $N_a$, the number of totally positive integers with fixed trace $a$ in a totally real field. That number has a natural geometric approximation $G_a$, and we ...
11
votes
2answers
499 views

For what real $t$ is $\{n^t : n \geq 1\}$ linearly independent over $\mathbb{Q}$?

It's straightforward that $t$ must be irrational. I have googled many variations of this question and browsed through some books on transcendental number theory. There is much that is said about when ...
7
votes
2answers
192 views

approximate two different real numbers to order $\frac{1}{z^{3/2}}$

I took this result from Minkowski's book on Geometry of numbers: Two arbitrary real quantitites $a$ and $b$ may be made to approach as near as we wish in value the two fractions $\frac{x}{z}$ and ...
13
votes
1answer
372 views

Small quotients of smooth numbers

Assume that $N=2^k$, and let $\{n_1, \dots, n_N\}$ denote the set of square-free positive integers which are generated by the first $k$ primes, sorted in increasing order. Question: what is a good ...
0
votes
0answers
33 views

How can vectors of a specific Diophantine type be computed?

We say that $\alpha \in \mathbb{R}^n$ is Diophantine of type $\kappa$ if there exists a constant $C$ such that \begin{equation} \max_{i \in \{1,\dots,n\}} | \, \alpha_i - \tfrac{m_i}{q} | \, > ...
2
votes
0answers
108 views

Linear forms with best approximation vectors lying in a subspace

Setup: For $u \in \mathbb{R}^n$, let $\rho(u)$ be the Euclidean length, $\sqrt{u_1^2 + \ldots + u_n^2}$. For $x \in \mathbb{R}$ let $\|x\| = \min_{k \in \mathbb{Z}} |x - k|$, and for $x \in ...
2
votes
0answers
116 views

A question related to metric Diophantine approximation

In metric Diophantine approximation you are often interested in finding conditions on $(\phi(q))_{q \geq 1}$ which guarantee that $$ \left| \alpha - \frac{p}{q} \right| < \frac{\phi(q)}{q} $$ has ...
0
votes
2answers
131 views

Comparing the Rational Approximability of Infinite Continued Fractions

It is known, that $\phi := \frac{sqrt(5)-1}{2}$, is the number, that is hardest to approximate by rationals (cf e.g. the section properties of the golden ratio $\phi$ here: ...
28
votes
4answers
3k views

Massive cancellations

Let $A=\{a_1,\ldots,a_k\}$ be a fixed, finite set of reals. Let $S_A(n)$ be the set of all reals that are expressible as the sum of at most $2^n$ terms, where each term is a product of at most $n$ ...
14
votes
1answer
361 views

Is there a bounded sequence of points in the plane with pairwise distances at least $1/\sqrt{|i-j|}$?

Previously I have mentioned the following problem in an addition to the list of Contest problems with connections to deeper mathematics.. Is there an infinite bounded sequence $(P_n) \subset ...
2
votes
2answers
368 views

Rate of convergence of an irrational rotation

Let $\alpha, \beta \in \mathbb{R}$. Let $\{x\}$ denote the fractional part of $x$ and let $\|x\| = \min(\{x\}, 1-\{x\})$. If we assume that $\alpha$ is irrational, then there exists an increasing ...
2
votes
1answer
214 views

Computing all “suboptimal” rational approximations to $\pi/2$

I have an irrational number $\alpha$ ($\alpha=\frac\pi2$), and I would like to determine all integers $n\in[1,N]$ ($N=10^{16}$) that satisfy $$ n \epsilon(n)^2 \leq \tau $$ where $\tau$ is a known ...
6
votes
0answers
344 views

Is simultaneous diophantine approximation (in a weaker sense) NP hard?

The traditional problem of simultaneous diophantine approximation is: Given a set of rational numbers $g_1,\ldots,g_d$, an integer $N$, and a rational $\gamma>0$, is there an integer $W$ with ...
6
votes
0answers
79 views

Diophantine approximation in $\mathbb{G}_m^r$ with approximants restricted to a finiteley generated subgroup

Faltings, in the same paper (Diophantine approximation on abelian varieties, Ann. Math. 1991) in which he proved his famous ``big theorem,'' proved also that at any place $v$ of a number field $K$ and ...
4
votes
1answer
146 views

Inequality due to Siegel (assumptions) and upper bounds on number field discriminants

In Siegel's 1969 paper, Abschätzung von Einheiten, on page 73, he states the inequality $$\log\sqrt d\le n-1+{n\over 2}\log\pi+r_2\log 2\qquad (*)$$ and compares with the bound due to Minkowski that ...
3
votes
0answers
94 views

Conjectural growth rate for ergodic sums of logarithms

Let $\theta, \phi \in [0,1)$, and consider the sums $$ S_n(\theta,\phi)=\sum_{k=0}^n \log|e^{2\pi i (k\theta+\phi)}-1|. $$ The possible boundedness from above of such sums plays a key role in ...
4
votes
1answer
183 views

Approximating rational values in ]0,1[ by a sum or difference of unit fractions

Let $U=\{\frac{1}{n}: n\in\mathbb{N}\} \cup \{-\frac{1}{n}: n\in\mathbb{N}\}$ be the set of positive and negative unit fractions. Are there positive integers $m<n \in \mathbb{N}$, such that for ...
4
votes
1answer
420 views

Lower bound on the irrationality measure of $\pi$

There seems to be a lot of work on the upper bound for the irrationality measure of $\pi$, but I could not find anything on a lower bound except the general $\mu(\pi)\geq2$. Looks like it is the best ...
18
votes
1answer
945 views

Irrationality measure of log(2)/log(6)

As part of my Phd thesis on aperiodic Wang tilings, I've discovered I need a bound on the irrationality measure of $\gamma = \log 2/\log 6$. That is, I am looking for an upper bound on the quantity ...
3
votes
1answer
178 views

Higher dimensional analogue of Thue's equation

The classical Thue equation is $$\displaystyle F(x,y) = h,$$ for a binary form $F(x,y) \in \mathbb{Z}[x,y]$. Recall that a binary form is a polynomial in two variables which is homogeneous, and $h$ ...
3
votes
0answers
194 views

Logarithms of ratios of squarefree numbers

Let $M \geq 1$, and $N=2^M$. Let $a_1, \dots, a_N$ be the set of all the numbers that you get when forming all square-free products of the first $M$ primes. For example, for $M=2$ and $N=4$ you get ...
0
votes
1answer
166 views

Reference request: on sums of the form $ax^m + by^n = h$

I know that equations of the form $$\displaystyle ax^d + by^d = h$$ with $a,b,h \in \mathbb{Z}$ have been thoroughly investigated as a special (and interesting) case of the Thue-Mahler equation, for ...
4
votes
1answer
171 views

Hermite Lindemann and transcendental reals

This is about compass and straightedge constructions, although I suspect nothing changes if we expand to real numbers algebraic over the rationals. Anyway, let $E$ be the "constructible numbers," ...
9
votes
1answer
335 views

Distribution of polynomials mod 1 using co-prime integers

If $a$ is any real irrational, then the set of numbers of the form $ax+y$ with $x$ and $y$ co-prime integers is dense in $\mathbb{R}$. I managed to prove this, in what I suspect is an overly ...
3
votes
1answer
181 views

Numbers with balanced diophantine approximations

This is a follow-up to Question 146635, namely Expected symmetry in the diophantine approximations of an irrational number, which I will refer to for notation and terminology used here without ...
7
votes
1answer
207 views

Expected symmetry in the diophantine approximations of an irrational number

Given $x \in \mathbb{R}$ we will write $\{x\}$ for the fractional part of $x$ and $\|x\|$ for the distance of $x$ from the nearest integer, in such a way that $\{x\} = x - \lfloor x \rfloor$ and ...
0
votes
0answers
170 views

On the irrationality measure of generalized Stoneham numbers

Pick non-zero integers $a,b,c$ with $a,b \ge 2$ and let $\xi_{a,b,c}$ be the sum of the series $\sum_{n=1}^\infty a^{-b^n} c^{-n}$ (no restriction is made on the sign of $c$); when $b = c$ and ...
8
votes
1answer
397 views

On the irrationality measure of $\sum_{n=1}^\infty a^{-b^n}$

Pick integers $a, b \ge 2$ and let $\xi_{a,b}$ be the sum of the series $\sum_{n=1}^\infty a^{-b^n}$. It is known that $\xi_{2,2}$ is transcendental: I learned a proof of this from notes by M. ...
2
votes
1answer
260 views

(efficient) method to test $\{n\alpha\}\not\in [A, B]\subset [0,1]$

Suppose $\alpha$ is a fixed given irrational number with $\alpha\in [A, B]\subset [0,1]$, are there any (efficient) methods to compute the least integer $n$ such that the decimal part of $n\alpha$ ...
2
votes
1answer
182 views

Height on a semiabelian variety

Let $A$ be a semiabelian variety over $\bar{\mathbb{Q}}$ and $B$ a semiabelian subvariety of $A$. Let $\pi:A\to A/B$ be the canonical morphism. Let $h:A(\bar{\mathbb{Q}})\to\mathbb{R}$ and ...
0
votes
0answers
145 views

Invariance of the (Liouville-Roth) irrationality measure under rational Möbius transformations

For a real number $x$, we define the (Liouville-Roth) irrationality measure of $x$, here denoted by $\mu(x)$, as the infimum, with respect to the poset $(\mathbb{R}_0^+ \cup \{\infty\}, \le)$, of the ...
10
votes
1answer
391 views

Distribution mod 1 of exponential growth sequences

Let $t_n$ be a sequence of real numbers and $C,r>1.$ Suppose that for every $n\geq 1$ we have $\frac{1}{C}r^n\leq t_n \leq Cr^n.$ Does there exist a real number $\xi$ and an $\varepsilon>0$ ...
8
votes
1answer
352 views

Do simple, multi-dimensional generalizations of this continued fraction formula exist?

Background Let $x\in [0,1)\setminus \mathbb{Q}$ have regular continued fraction expansion given by $$ x = [a_1, a_2, a_3, \dots] = \cfrac{1}{a_1+\cfrac{1}{a_2+\dots}}, \qquad a_i \in \mathbb{N}. $$ ...
1
vote
0answers
150 views

Dalzel's integral for $\pi$ and the lemniscate constant

$\pi=\varpi_2$ can be considered as a member of the sequence of real numbers $$\varpi_m=2\int\limits_0^1\frac{dt}{\sqrt{1-t^m}},$$ and, for example, the Wallis product formula ...
3
votes
0answers
127 views

Diophantine approximations by norms of quadratic irrrationalities

The following problem came up on a mailing list that I subscribe to: If $\alpha$ is irrational we can find (using continued fractions) infinitely many rational fractions $p/q$ such that $|q \alpha - ...
5
votes
3answers
395 views

Siegel's theorem with real coefficients

Let $h(x,y)$ be a polynomial with real coefficients. Suppose there are infinitely many integer solutions to $|h(x,y)|<1$. What can I say about $h$? When $h$ itself has integer coefficients, a ...
8
votes
3answers
618 views

Is the infimum of Salem numbers > 1?

BACKGROUND A Salem number is an algebraic integer $\theta$ such that all the Galois conjugates of $\theta$ are $\leq 1$ in absolute value, and at least one of them lies on the unit circle. Their ...
4
votes
1answer
378 views

Winning sets of full measure (Schmidt's game)

A quick reminder of the definition of Schmidt's game: Let ${X}$ be a metric space and ${S\subset X}$ be a subset. Let ${0<\alpha,\beta<1}$ be constants. Bob chooses any open ball ...
1
vote
1answer
197 views

Approximation by binary fractions

For general Diophantine approximation, the Thue–Siegel–Roth theorem states that for any irrational algebraic number $x$, and any $\varepsilon>0$, there exists a constant $c=c(x,\varepsilon)$ such ...