# Tag Info

### Existence of a positive measurable set with disjoint preimage under iterated transformation

If $(X,\mathcal{B},\mu)$ is a Lebesgue probability space, only the aperiodicity is needed by a simple application of Rokhlin's lemma: There exists $B\in\mathcal{B}$ such that $B,\dots,T^{-k}(B)$ are ...
• 2,857

### Existence of a positive measurable set with disjoint preimage under iterated transformation

The statement is false in general, I added a counterexample at the end of my answer to show that some separability condition like countable separability (or the stronger condition of being Lebesgue ...
• 8,206
1 vote

### Entropy arguments used by Jean Bourgain

What follows is a bit of Computer-Sciency exposition, but since it proves the desired claim and it is an "elementary entropy consideration" I hope it answers the question. For convenience, ...
1 vote

### Strict positive definite function gradient tuple

We need to show for all $a\in \mathbb{R}^n$ and all $v_1,\dots, v_n\in \mathbb{R}^d$ $$\text{Var}\Bigl[\sum_{k=1}^n a_k f(x_k) + D_{v_k}f(x_k)\Bigr] > 0$$ We assume $f$ to be centered by ...
• 367

### Fourier series of smooth functions in infinitely many variables

Not sure why this is brought up after some years. But I think you should google "Fourier analysis on infinite torus" or something similar. The first result returns me a paper by Denis Fufaev....
• 926

• 103k
1 vote
Accepted

### Rate of convergence of the minimum point over a product space

$\newcommand\ep\epsilon\newcommand\th\theta\newcommand\de\delta$The answer is yes, even with $C=1$ for all such $f$. Indeed, let $f$ satisfy all your conditions on $f$. Let $g(0):=0$. For real $x>0$...
• 118k
Accepted

• 118k

### Inverse Limit in the category of $C^{\ast}$-algebras or operator spaces

You can see in the paper "Some inverse limits of Cuntz algebras" when the inverse limits (Projective limits) exist in the category of C*-algebras.
Accepted

• 2,710
1 vote

### Is every face exposed if all extreme points are exposed?

It is also possible to construct counterexamples from the idea that every extreme point of the convex hull of any subset of the unit sphere $S^2$ is an exposed point. Placing a convex set $C$ inside ...
• 31
1 vote

• 56.6k
Accepted

• 5,163

### Functions with asymmetrically decreasing Fourier transform?

I suddenly realized that there is a simple explicit way of getting the function $f_1=F$ in Alexandre Eremenko's answer. Just for record, I put it here as an answer. The idea is to take as $g$ the ...
• 3,390
Accepted

### Can a lift satisfy Chen's relation, geometric condition but not be a rough path?

Indeed, this can fail. Take any geometric rough path lift $\mathbb X^{ij}$ and simply add in a non-symmetric way the increment of a function $F$ that fails to be $2\gamma$-Hölder. For example, take ...
• 4,449
Accepted

• 118k

### How to understand the unique continuation result

If I understand things correctly, $u$ is vanishing on some non-empty open subset. Moreover, you have pointwisely on $\mathbb R^3$ the differential inequality  \lvert \Delta u\rvert\le C\lvert u\...
• 15.2k

### Combination of simple tensors - II

Take $X = Y= M_2$ and consider the tensor $e_1 \otimes e_1 + e_2 \otimes e_2$. You get the same result for any choice of orthonormal basis of $\mathbb{C}^2$, so there can already be two expressions ...
• 42.2k

### Simple proof that exactness implies strong mixing

I think the martingale theory really captures exactly what's happening. I don't know any other proof, but suspect that you would end up reproducing the backwards martingale theorem in some form? I am ...
• 22.5k