5 problems

Let $k$ be a positive integer and let $S\subset\mathbb{N}$ be chosen independently at random with … Here, $f=o(g)$ if and only if $g=\omega(f)$. Frantzikinakis–Lesigne–Weirdl conje…

Let $P_1,\dots,P_s\in\mathbb{Z}[x]$ be linearly independent polynomials that vanish at zero. For $\omega\in\{0,1\}^s$, write $\omega=(\omega_1,\dots,\omega_s)$ and…

Let $P_1,\dots,P_s\in\mathbb{Z}[x]$ be linearly independent polynomials that vanish at zero. Let $f_\omega:\mathbb{Z}/N\mathbb{Z}\to\mathbb{C}$ be one-bounded functions for each…

Let $d\geq 1$, and let $\mathcal{P}$ denote the set of prime numbers. A subset $A\subseteq\mathcal{P}^d$ is dense if it has positive relative density in the $d$-tuples of primes. A…

Let $\mathcal{H}$ denote the Hardy field under consideration, let $[x]$ be the integer part of $x$, and let $\bar d(\Lambda)$ denote the upper density of $\Lambda\subset\mathbb{Z}$…