Open problems in Mathematics stemming from research papers

Cosine Sign Correlation

Let $\left\{a_1, \dots, a_n\right\} \subset \mathbb{N}$ be a set of $n$ distinct positive integers. Let $X$ be a uniformly distributed random variable on $[0,2\pi]$. How small can the probability ...
Open
Posted online: 2024-03-20 15:31:29Z by Stefan Steinerberger20
- Classical Analysis and ODEs
- Probability
The Size of Intervals in the Hardy-Littlewood Maximal Function

Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is a continuous function (more smoothness can be assumed if deemed necessary). A classical object is the Hardy-Littlewood maximal function $\mathcal{M}f: \mathbb{R} \rightarrow \mathbb{R}$ defined via $$ (\mathcal{M} f)(x) = \sup_{r >0} \frac{1}{2r} \int_{x-r}^{x+r} |f(z)| dz.$$ ...
- Stefan Steinerberger
Open
Posted online: 2024-03-20 15:17:20Z by Stefan Steinerberger20
- Classical Analysis and ODEs
Littlewood's Cosine Root Problem

Let $A \subset \mathbb{N}$ be a set of positive integers. One basic intuition is that the function

$$ f(x) = \sum_{k \in A} \cos(kx)$$

should have many roots on $[0, 2\pi]$ when the set $A$ is large. Littlewood conjectured that the number of roots is probably something like $\#A - 1$. This was disproven in 2008 by Borwein, Erdelyi, Ferguson, Lockhart (Annals). The best bounds are ...
- J. E. Littlewood
Open
Posted online: 2024-03-20 15:04:36Z by Stefan Steinerberger13
- Classical Analysis and ODEs
Symbolic dynamics for real Hénon and Lozi maps (By S. \v Stimac)

Kneading theory is a combinatorial tool to understand the dynamics of a piecewise monotone map from the interval to itself and was developed by Milnor and Thurston (Lecture Notes 1988).

The set of kneading sequences, denoted as $\mathfrak{K}$, consists of all sequences $\bar{k}^n$ for $n \in \mathbb{Z}$. Each kneading sequence $\bar{k} = \bar{k}^n$, where $n \in \mathbb{Z}$, follows the structure: ...
- Eric Bedford,
- Pierre Berger,
- Fabrizio Bianchi,
- Xavier Buff,
- Sylvain Crovisier,
- Tien-Cuong Dinh,
- Romain Dujardin,
- Charles Favre,
- Tanya Firsova,
- Patrick Ingram,
- Yutaka Ishii,
- Liviana Palmisano,
- Enrique Pujals,
- Jasmin Raissy,
- Sonja Štimac,
- Gabriel Vigny
Open
Posted online: 2024-03-18 19:41:03Z by SciLag Admin14

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- Dynamical Systems
Dissipative real Hénon maps (by S. Crovisier and E. Pujals)

Consider the quadratic real Hénon maps $$f_{c, b} (x, y):= (x^2+c-by, x).$$

The map $f_{c,b}$ is called mildly dissipative if $|b|< 1$ and if for any ergodic invariant measure $\mu$, not supported on a sink, and for $\mu$-almost every point $x$, both components of the stable curve $W^s(x)\setminus \{x\}$ are unbounded. ...
- Eric Bedford,
- Pierre Berger,
- Fabrizio Bianchi,
- Xavier Buff,
- Sylvain Crovisier,
- Tien-Cuong Dinh,
- Romain Dujardin,
- Charles Favre,
- Tanya Firsova,
- Patrick Ingram,
- Yutaka Ishii,
- Liviana Palmisano,
- Enrique Pujals,
- Jasmin Raissy,
- Sonja Štimac,
- Gabriel Vigny
Open
Posted online: 2024-03-18 19:41:03Z by SciLag Admin16

View the group
- Dynamical Systems
Real Hénon maps (by P. Berger)

Question 1: Can we find a Hénon map (of some degree) which admits a wild attractor, i.e., a Cantor set which attracts a set of Lebesgue positive measure and which is strictly included in a transitive set ?

Refer to Van Strien (2010) who asked whether a suitable analog of wild attractors could exist for Hénon maps (of some degree). ...
- Eric Bedford,
- Pierre Berger,
- Fabrizio Bianchi,
- Xavier Buff,
- Sylvain Crovisier,
- Tien-Cuong Dinh,
- Romain Dujardin,
- Charles Favre,
- Tanya Firsova,
- Patrick Ingram,
- Yutaka Ishii,
- Liviana Palmisano,
- Enrique Pujals,
- Jasmin Raissy,
- Sonja Štimac,
- Gabriel Vigny
Open
Posted online: 2024-03-18 19:41:03Z by SciLag Admin7

View the group
- Dynamical Systems
Collatz Problem

The Collatz conjecture is among the most renowned unsolved problems in mathematics. It postulates whether iteratively applying two elementary arithmetic operations will inevitably lead every positive integer to 1.

It revolves around sequences of integers where each term is derived from the preceding one by the following rules: ...
- E.F.,
- O'Connor,
- J.J.; Robertson
Open
Posted online: 2024-03-18 07:48:00Z by SciLag Admin26
- Number Theory

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