15
votes
Accepted
Upper bound for Hall's conjecture on separation of squares and cubes
The best $\theta$ is $0$. It is known that there are infinitely many
solutions of 0 < $|x^3 - y^2| \ll x^{1/2}$, parametrized by certain
"Pell equations"; indeed one such family attains $...
- 77.5k
13
votes
Accepted
Continuous variant of the Chinese remainder theorem
Your guess is correct. For $\alpha = (\alpha_1, \alpha_2, \ldots, \alpha_d) \in \mathbb{R}^d$, the values of $m \alpha \bmod 1$ are equidistributed (and in particular dense) in $(\mathbb{R}/\mathbb{Z})...
- 151k
10
votes
Accepted
Integral points near elliptic curves
You can take $\theta = 0$, even $\theta = -1/6$ works.
Fix an integer $A \ne 0$ and an integer $B$. If $r$ is an integer, the elliptic curve $E : y^{2} = x^{3} + Ax + r^{2} A^{2}$ has the obvious ...
- 20k
7
votes
Accepted
Central limit theorem for irrational rotations
I am surprised by the question. The sums are bounded if $\alpha \ne 1$, since for every $n \ge 1$,
$$\Big|\sum_{k=1}^n \Re R_\alpha^k z \Big|
= \Big|\Re \Big(\sum_{k=1}^n R_\alpha^k z \Big) \Big| = \...
- 3,617
6
votes
Accepted
The liminf of an expression involving an irrational rotation
The answer is no, because for some real $c>0$ and all integers $q>0$ we have
$$q\{q\sqrt2\}=q(q\sqrt2-\lfloor q\sqrt2\rfloor)
=q|q\sqrt2-\lfloor q\sqrt2\rfloor| \\
\ge q\,\inf_{p\in\mathbb Z}|q\...
- 118k
4
votes
Accepted
Diophantine equations involving recurrence sequences
The authors skipped some easy steps. By the triangle inequality,
$$e^{z_1}\geq 1-|1-e^{z_1}|>1-0.95=1/20,$$
whence using also $z_1<0$,
$$e^{|z_1|}=e^{-z_1}<20.$$
Similarly,
$$e^{z_2}\geq 1-|1-...
- 99.2k
3
votes
Accepted
Simultaneously approximating all $x \in [0,1]$ with fractions of bounded denominator
Fix $T$ ($T=100$ in your example) and denote by $S_T(n)$ the set of numbers $x\in [0,1]$ such that $|x-p/q|>1/n^2$ whenever $p,q$ are integers and $1\leqslant q\leqslant Tn$ (so, my $n$ is your $\...
- 103k
3
votes
Accepted
Simultaneous rational approximations of multiples of the golden ratio
Note that since you want $|n_m \alpha| \to 0$ this implies that for almost all $m$ you have $|n_m \alpha| < \frac{1}{2k}$, and in that case you have $ |n_m k \alpha | = k |n_m \alpha |$. You can ...
- 2,979
3
votes
Accepted
Extreme case bounds on Diophantine approximation
If $\alpha$ is a real irrational number, then there are infinitely many coprime integers $p,q$ with $q > 0$ such that
$$\displaystyle \left \lvert \alpha - \frac{p}{q} \right \rvert < \frac{1}{q^...
- 25.5k
1
vote
Diophantine equations involving the difference between perfect square and perfect cube
Just to point out that this is related to integral points
on elliptic curves, which is well studied topic.
Treating $z$ as parameter, your equation (a) is equivalent to
integral points on the
elliptic ...
- 24.2k
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