User kale - MathOverflow

Hypergeometric method

2012-06-21T21:05:12Z

Suppose I have an approximation $p_0/q_0$ to an irrational number $\alpha$. The hypergeometric method uses the fact that if $|Q_n\log-P_n| < C*b^{-n}$ and $|Q_n| < a^n$ then $log a/log b$ is the exponent of diophantine approximation. I'm wondering how I can determine when this is an immprovement on liouville's inequality. I know it clearly depends on the an and b, so for my question I'm looking for a and b to be given explicitly and then a description of how I might be able to determine when loga/logb is an improvement on liouville's inequality.

how do you bound exponent of x^2+1=y^p

2012-06-15T19:22:20Z

for p a prime exponent using linear forms in logs?

So far I have (x-i)(x+i)=y^p which are coprime and hence x+i=(a+ib)^p , now how do I get a linear form in logs so that I can find an upper bound on p?

Folklore Lemma (p-adic case)

2012-05-31T01:24:00Z

Is there a p-adic version of the so called 'folklore lemma' which relates a sequence of diophnatine approximations to an algebraic number to the exponential of irrationality.

Sharpenings of Liouville's inequality

2012-05-30T22:12:51Z

The norm of an algebraic number $\alpha$ is the product of its conjugates, $N(\alpha)$.

Suppose that I have an inequality of the form $|x-\alpha*y| > c X^{n-\gamma}$ where $X=max{|x|,|y|}$ and c is some constnat, can this be used to find an upper bound on the norm of $x-\alpha*y$ of the form $|N(x-\alpha*y)| < C *X^{n-\gamma}$ $|x-\alpha*y|$ where C is some constant and $\gamma \geq 1$

In the case of $\gamma=1$ it can since: $|N(x-\alpha*y)|=\prod (x-\alpha_i*y)|=|x-\alpha*y|X^{n-1} \prod (x/X-\alpha_i*y/X) \leq |x-\alpha*y| \prod(1|+|\alpha_{i}|)X^{n-1}$

It's unclear to me how it might be possible to find a sharper upper bound on the norm of $x-\alpha*y$ by using a sharper exponent in liouiville-type inequalities.

I should add that the constant C should be effective, as it is in the case of $\gamma=1$.

p-adic approximations to hypergeometric functions

2012-05-17T18:26:48Z

I'm trying to research this question and I need links to any of the following papers.

"Baker A (1964) Rational Approximation to certain algebraic numbers. Proc. London Math. Soc 4: 385-308"

"Alladi K and Robinson ML (1980) Legendre polynomials and irrationality. K. Reine Angew. Math 318: 137-155"

I can't find any of them online though.

computing the measure of a generator in a fixed number field K

2012-04-28T19:53:19Z

Suppose $K=\mathbb{Q}(\alpha)$ is a fixed number field with $[K:\mathbb{Q}]=d$ and fixed basis $b_1,b_2,..,b_d$. Define $$m(\alpha) = max \{ | p_i| , |q_i| : 1 \leq i \leq d \},$$ where the max is taken over all representatives of $\alpha$ of the form $$\alpha = \frac{p_1b_1+p_2b_2+\ldots+p_db_d}{q_1b_1+q_2b_2+\ldots+q_db_d}.$$

Are there any papers which describe techniques to compute this number?

The p-adic subspace Theorem

2012-05-06T21:02:45Z

Could someone explain how the subspace theorem could be used to transfer results from archimedian valuatons to nonarchimedian ones?

p-adic version of Liouville's approximation theorem

2012-05-05T05:57:56Z

Does anyone know of a p-adic analogue of Liouville's approximation theorem http://mathworld.wolfram.com/LiouvillesApproximationTheorem.html with proof?

I'm aware of Roth's theorem and subspace theorems which can be generalized to p-adic numbers, these are different in that the constants are not effective.

p-adic diophantine approximation

2012-05-04T05:31:27Z

Suppose you have a sequence of rational numbers that gives a diophantine approximaion an irrational, what can be said p-adically about this sequence?

I'm interested in the p-adic analoges of these theorems (such as Thue-siegel-roth), but can't find any straightforward resources on the subject. I can't even find what a good diophantine approximation would mean over a p-adic field.

p-adic analysis of hypergeometric functions

2012-05-02T22:57:28Z

Are there any p-adic techniques that can be applied to the 2F1 hypergeometric function?

For e.g. I'm interested in which values this function converges p-adically.

Lower Bounds on Binary Forms of degree m

2012-04-30T20:43:42Z

The following paper http://www.math.leidenuniv.nl/~evertse/05-discres.pdf looks at the binary form $\sum \alpha_ix^iy^{m-i} where i ranges from 0 to m. It then defines the discriminant.

It's well known that for binary quadratic forms how to find a lower bound using the discriminant. I was wondering if analogous theorems hold for a form of arbitrary degree.

Ultimately I would like to find a lower bound in terms of the variable y for this binary form I'm working on.

A little more details:

I'm trying to show that the binary form $x^{n-1}+x^{n-2}y\alpha^2+x^{n-3}y^2\alpha^3+...+y^{n-1}\alpha^{n-1}$ is bounded below by $c*y^{n-1}$ or something similar where c is some explicit constant.

Is it true that if X is a separable banach space and M a inear manifold in the dual

2012-04-15T19:29:10Z

that weak sequential closure M = weak star closure then M is sequentially dense?

A point in the weak closure but not in the weak sequential closure

2012-03-14T20:36:46Z

I'm trying to find a proof of this counterexample by von Neumann:

Let $x_{mn}\in \ell^2$ be defined by $$x_{mn}(m)=n \quad,\quad x_{mn}(n)=m \quad,\quad x_{mn}(k)=0 \hbox{ otherwise,} $$ and let $S=\{ x_{mn} : m, n\geq 1\}$ . Von Neumann shows that $0$ is in the weak closure of this set but no sequence in $S$ convergess weakly to $0$.

Comment by Kale

2012-06-15T20:34:52Z

not what link is supposed to mean, I'm not asking for a full proof of cataln's conjecture but consider this particular e.g.

Comment by Kale

2012-05-30T22:44:04Z

Gerry Myserson: I'm not sure what is stronger/weaker here. I would like to use a stronger Liouiville's inequality to imply an upper bound of a certain form on the norm of the algebraic number $x-\alpha*y$ (in absolute value). Basically I want $\prod (x-\alpha_i*y) (where i ranges over all conjugates of $\alpha$ other than itself to be $ \leq C*X^(\gamma=1)$ given that the stronger Liouville inequality $|x-\alpha*y| \leq c *X^{\gamma-1}$ does hold.

Comment by Kale

2012-05-30T22:23:30Z

To make my questions a bit more clear: In the case of $\gamma=1$, the inequality follows immediately, I would like a proof that works for any $\gamma$, given the lower bound on the absolute value of $x-\alpha*y|$ I think there should be some way to use a lower bound on the absolute value of an algebraic number and transform it into an upper bound on the norm as is the case for $\gamma=1$.

Comment by Kale

2012-05-07T00:16:43Z

I was just looking for an e.g. of how to apply it

Comment by Kale

2012-05-05T08:51:25Z

I'm intereste in ways that ordinary Liouville's theorem relates to p-adic Liouville's theorem

Comment by Kale

2012-05-05T08:33:20Z

Anatoly: Is this the only paper on the subject? It looks like it's generalized over any extension field.

Comment by Kale

2012-05-04T06:26:53Z

I'm aware of these particular references listed on the wikipedia page, the problem is I can't find them anywhere.

Comment by Kale

2012-04-15T20:40:26Z

Op here. M is just a linear subspace of X^*. Spaces where weak sequential closure is the same as ordinary closure (e.g. first countable spaces) are called Uryshon Fretchet spaces. Ive seen various characterizations of when M is sequentially dense in the dual of a separable space but none of them are very helpful. I'm trying to determine under what conditiosn M is a Uryshon Fretchet space.