• The Erdos Distinct Subset Sum Problem

Let $a_1 \leq a_2 \leq \dots \leq a_n$ be a set of $n$ positive integers such that all $2^n$ subsets of the set are uniquely identified by the sum of their elements. An example of such a sequence are the powers of 2 (this is the uniqueness of the binary expansion in disguise). The question, due to Erdos in 1931 or 1932, is whether sets with such a property have to always have a large number. ...

Open

Posted online: 2024-03-22 16:13:14Z by Stefan Steinerberger41

• Combinatorics
• The Inverse of the Star Discrepancy

Suppose we have a finite set of points $\mathcal{P} \subset [0,1]^d$ with the following nice equidistribution property: for any axis-parallel rectangle $R \subset [0,1]^d$ with one point anchored in the origin, the number of points from $\mathcal{P}$ that are contained in the rectangle $R$ is very nearly proportional to the volume of the rectangle ...

Open

Posted online: 2024-03-22 15:58:04Z by Stefan Steinerberger15

• Combinatorics
• Create convexity in 3 (or 100) steps only (M. Talagrand)

Create convexity in 3 (or 100?) steps only! Consider an integer $N$. Let us say that a compact subset $A$ of $\mathbb{R}^N$ is balanced if $x \in A$, $\lambda \in \mathbb{R}$, $|\lambda| \leq 1 \Rightarrow \lambda x \in A$. Let us denote by $\gamma_N$ the canonical Gaussian measure on $\mathbb{R}^N$. ...

Unconfirmed

Posted online: 2024-03-21 15:39:31Z by SciLag Admin27

• Combinatorics
• Make 1000 USD from simple combinatorics (M. Talagrand)

Make 1000 USD from simple combinatorics. This is Research Problem 12.5.6 of the book Upper and Lower Bounds. You have to have a look at the book to get the connection between this problem and stochastic processes, but the problem can be understood without knowing anything about this book. ...

Open

Posted online: 2024-03-21 13:07:56Z by SciLag Admin51

• Combinatorics
• Cube always hitting a lattice point

Is there a universal constant $c$ such that all translation and rotations of a cube $Q \subset \mathbb{R}^n$ with side-length $c$ always contains a point from the $\mathbb{Z}^n$? ...

Open

Posted online: 2024-03-20 15:50:20Z by Stefan Steinerberger15

• Combinatorics
• Classical Analysis and ODEs
• 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 Steinerberger14

• 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 $$(\mathcal{M} f)(x) = \sup_{r >0} \frac{1}{2r} \int_{x-r}^{x+r} |f(z)| dz.$$ ...

Open

Posted online: 2024-03-20 15:17:20Z by Stefan Steinerberger17

• Classical Analysis and ODEs
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