12 problems

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 $v_1,\ldots,v_n$ 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…

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…

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 $S$.

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…

Singularity probability conjecture. The singularity probability satisfies

Let $x_1,\dots,x_n$ be independent and uniformly chosen from $\{-1,1\}$, and let … be a degree-$k$ homogeneous polynomial with at least $mn^{k-1}$ nonzero coefficients. Polynomial…

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…