24 problems

Let $v_1,\ldots,v_n$ be unit vectors in $\mathbb{R}^2$, where $n$ is odd, and let $\epsilon_1,\ldots,\epsilon_n$ be independent Rademacher random variables. Odd-$n$ Erdős conjectur…

Erdős–Moser conjecture. The logarithmic factor in the bound ${\mathbb P}_1({\mathbf v})=O(n^{-3/2}\ln n)$ is unnecessary; in particular, one should have…

Let $n$ be a positive integer, let $v_1,\dotsc,v_n\in\mathbb{R}^2$ be unit vectors, and let $\xi_1,\dotsc,\xi_n$ be independent random variables uniformly distributed on…

Let $S\subseteq\mathbb{R}^k$ be a set of points in convex position, meaning that no point of $S$ can be represented as a convex combination of the others, and let $a_1,\ldots,a_n$…

Let $a_1,\ldots,a_n\in\mathbb{C}^k$ be vectors from which one can form at least $b$ pairwise disjoint bases of $\mathbb{C}^k$. Let $S\subseteq\mathbb{C}^k$ be an affine algebraic v…

Let $\xi_1,\ldots,\xi_n$ be independent Rademacher random variables, and let $F$ be an $n$-variable polynomial of degree $d$. Suppose that $F$ has at least $bn^{d-1}$ nonzero coeff…

Let $V\subseteq\mathbb{R}^2$ be a set of $n$ unit vectors, and let $\sigma_V$ denote the corresponding random signed sum. He–Juškevičius–Narayanan–Spiro conjecture. For all suffici…

Let $v_1,\dotsc,v_n\in\mathbb{R}^2$ be unit vectors, and let $\varepsilon_1,\dotsc,\varepsilon_n$ be independent random signs, each uniformly distributed in $\{-1,1\}$. He–Juškevič…

Let $x_1,\dotsc,x_n$ be unit complex numbers, and let $\varepsilon_1,\dotsc,\varepsilon_n\in\{-1,+1\}$. Erdős's conjecture. The number of sums … with … is greater than $c2^n/n$ for…

For each sufficiently large integer $n$, let $v_1,\ldots,v_n$ be arbitrary unit vectors in $\mathbb{R}^2$, and let $\epsilon_1,\ldots,\epsilon_n$ be independent Rademacher random v…

Let $0\le d<r$ and $q$ be integers. Let $\mathcal{Z}\subseteq\mathbb{R}^{r}$ be an algebraic variety of dimension $d$ and degree at most $q$. Let…

Convex-position Littlewood–Offord conjecture. If $n$ is sufficiently large in terms of $\varepsilon$ and $d$, then

Optimal point-concentration conjecture. In the setting of Theorem baby-forwards, one has $\Pr(X\in S)\le C_S/\sqrt n$ for some constant $C_S$ depending only on $S$.

Dzindzalieta–Juškevičius conjecture. For arbitrary norm $\\|\cdot\\|$ on $\mathbb{R}^d$, if $\mathbf{v}_i\in\mathbb{R}^d$ satisfy $\\|\mathbf{v}_i\\|\leq 1$ and…

Let $X_1,\ldots, X_n$ be iid random vectors in $\mathbb{R}^d$ satisfying … A choice of weights $w_i\in \\{-1,1\\}$ should exist such that, for all non-zero $a_i\in \mathbb{R}$ and…

Let $X_1,\ldots, X_n$ be iid random vectors in $\mathbb{Z}^d$. A choice of weights $w_i\in \\{-1,1\\}$ should exist such that, for all non-zero $a_i\in \mathbb{R}$ and all…

Let the non-uniform Littlewood–Offord theorem refer to Theorem, whose statement uses the Euclidean norm $\lVert\cdot\rVert_2$ on $\mathbb{R}^n$. Norm-invariance conjecture. Theorem…

Let $U_1,\ldots,U_n$ be independent uniform random variables on the arithmetic progression $\\{-m+1,-m+3,\ldots,m-3,m-1\\}$, where $m\geq 3$. Let $x\in\mathbb{R}^d$ and…

Let $G$ be any group and fix an odd integer $m\geq 3$. Suppose that all possible even orders of elements in $G$ greater than $m$ are given by the sequence…

Let $a_1,\ldots,a_n$ be distinct non-zero real numbers, and let $S_A$ denote their random Bernoulli sum. Erdős–Moser conjecture. The logarithmic factor in the bound … is not necess…

Singularity probability conjecture. The singularity probability satisfies

Let $k$ be a fixed positive integer. Let $y_1,\dots,y_k$ be independent vectors uniformly chosen from $\{-1,1\}^n$, let $A$ be a $k$-multilinear form whose coefficients are all non…

Let $A$ be an $n\times n$ matrix of nonzero entries, and let $x,y$ be chosen independently and uniformly from $\{-1,1\}^n$. For a matrix decomposition $A=A_1+A_2+A_3$, say that…

Let $\xi_1,\ldots,\xi_n$ be independent Bernoulli random variables, and let … be a quadratic form with non-zero coefficients $c_{ij}$. Quadratic Littlewood–Offord conjecture. For e…