Mathematics Problem Archive

Let $p_1,\ldots, p_\ell$ be integer valued generalized polynomials. Show that the averages $$ \frac{1}{N}\sum_{n=1}^{N} T_1^{p_1(n)}f_1\cdots T_\ell^{...

Suppose that the polynomials $p_1,\ldots, p_\ell\in \mathbb Z[t]$ are pairwise independent. Show that there exists $d\in \mathbb N$ such that the fact...

Suppose that the polynomials $p_1,\ldots, p_\ell\in \mathbb Z[t]$ are rationally independent. Show that $\mathcal{K}_{rat}(T_1),\ldots, \mathcal{K}_{r...

Suppose that the polynomials $p_1,\ldots,p_\ell\in \mathbb Z[t]$ are rationally independent and have zero constant term. Show that for every $A\in \ma...

Let $(X,\mathcal X,\mu,T_1,\ldots, T_\ell)$ be a system and $\{p_1,\ldots, p_\ell\}$ be a family of intersective integer polynomials. Show that for ev...

Let $(X,\mathcal X,\mu, T,S)$ be a system and $f,g\in L^\infty(\mu)$ be functions. Show that the averages $$ \frac1N \sum_{n=1}^N f(T^nx)\cdot g(S^nx)...

Is it true that one always has a decomposition $$ \int f_0 \cdot T_1^n f_1 \cdot \ldots \cdot T_\ell^n f_\ell \ d\mu= \psi(n)+e(n) $$ where $(\psi(n))...

Let $(X,\mathcal X,\mu)$ be a probability space, $T_1,\ldots, T_\ell \colon X\to X$ be invertible measure preserving transformations, and $p_1,\ldots,...

Here $\mathcal F=\{a_1,\ldots,a_\ell\}$ is a family of functions of polynomial growth in one Hardy field, and $\operatorname{span}^*(\mathcal F)$ deno...

Here $\mathcal F=\{a_1,\ldots,a_\ell\}$ is a family of functions of polynomial growth in one Hardy field, and $\operatorname{span}^*(\mathcal F)$ deno...

Let $a,b$ be distinct positive non-integers. Show that for every ergodic system $(X,\mathcal{X},\mu,T)$ and functions $f, g \in L^\infty(\mu)$, we hav...

Here $\mathcal F=\{a_1,\ldots,a_\ell\}$ is a family of functions of polynomial growth in one Hardy field, and $\operatorname{span}^*(\mathcal F)$ deno...

Find an example of a function $a\in \mathcal H$ that grows faster than polynomials, meaning, $a(t)/t^k \to\infty$ for every $k\in \mathbb N$, such tha...

Let $c$ be a positive non-integer. Show that the sequence $([p_n^c])$ is good for multiple recurrence and convergence of powers....

Show that the sequence $([n \sin n])$ is good for multiple recurrence and convergence of powers....

Show that if $c>1$ is not an integer, then the sequence $([n^c])$ is good for multiple recurrence and convergence of commuting transformations. Moreov...

Let $\ell \in \mathbb N$ and $c,c_1,\ldots, c_\ell$ be positive real numbers. Show that the prime numbers contain patterns of the form $$ \{m,m+[n^{c}...

Suppose that $n\sigma_n\to\infty$. Show that almost surely the sequence $(a_n(\omega))$ is good for multiple recurrence and convergence of commuting t...

Suppose that $n\sigma_n\to\infty$. Show that almost surely the following holds: For every system $(X,\mathcal X,\mu, T,S)$ and functions $f, g \in L^\...

Suppose that $a,b\in(0,1)$ and $a\neq b$. Show that almost surely the following holds: For every system $(X,\mathcal X,\mu,T,S)$ and functions $f, g \...

Let $(X,\mathcal X,\mu, T_n)$ be a measure preserving system with multiplicative structure and $A\in \mathcal X$ with $\mu(A)>0$. Is it true that ther...

Let $(X,\mathcal X,\mu, T_n)$ be a measure preserving system with multiplicative structure and $A\in \mathcal X$ with $\mu(A)>0$. Is it true that ther...

For a Hamiltonian diffeomorphism of a closed symplectic manifold, prove the Arnold lower bound on its number of fixed points in terms of Morse-theoret...

For a generic quantum system whose classical counterpart is integrable, prove that the unfolded high-energy level spacings have Poisson statistics....

Does there exist an ergodic measure-preserving transformation whose Koopman operator has simple Lebesgue spectrum?...

For a smooth dissipative dynamical system with a global attractor, is the supremum of the local Lyapunov dimension on the attractor attained at an equ...

Under the hypotheses in which the Lyapunov (Kaplan–Yorke) dimension is defined from the ordered Lyapunov exponents, prove that it equals the appropria...

Classify invariant ergodic probability measures for higher-rank diagonalizable group actions on homogeneous spaces; in particular, prove that the meas...

Prove that the outer billiard about almost every convex polygon has an unbounded orbit....

Let $M$ be a compact negatively curved Riemannian manifold. Do the probability measures $|\varphi_j|^2\,d\operatorname{vol}$ associated with every ort...

Is every strongly mixing measure-preserving transformation strongly mixing of order three?...

Starting from a positive integer $a_0$, define $a_{n+1}=\lfloor a_n^{1/2}\rfloor$ when $a_n$ is even and $a_{n+1}=\lfloor a_n^{3/2}\rfloor$ when $a_n$...

For which classes of ordinary differential equations do the classical and canonically generalized forms of Lyapunov's second method give necessary as ...

In every dimension at least three, is each reversible cellular automaton locally reversible?...

In a regular $n$-gon, call a trajectory preclosed when its endpoints divide their boundary edges into equal-length parts and meet those oriented edges...

Unfold a billiard trajectory in a regular $n$-gon by reflecting the polygon across each encountered side. A short trajectory joins two vertices and me...

Unfold a billiard trajectory in a regular $n$-gon by reflecting the polygon across each encountered side. A short trajectory joins two vertices and me...

Unfold a billiard trajectory in a regular $n$-gon by reflecting the polygon across each encountered side. A short trajectory joins two vertices and me...

Unfold a billiard trajectory in a regular $n$-gon by reflecting the polygon across each encountered side. A short trajectory joins two vertices and me...

Unfold a billiard trajectory in a regular $n$-gon by reflecting the polygon across each encountered side. A short trajectory joins two vertices and me...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...