5 problems

Let $X_0\in\mathbb N$ and $\varrho>0$, and let $X$ be sufficiently large in terms of $X_0$ and $\varrho$. Suppose that $A,B\subseteq [X]$ and that … for all primes…

Let $\mathcal A$ be a set of positive integers. For each prime $p$, let $\mathcal A_p$ denote the set of residue classes occupied by $\mathcal A$ modulo $p$. A rational quadratic i…

Let $N$ be a positive integer and let $A\subseteq [N]$. For each prime $p$, define the set of occupied residue classes … A rational quadratic is the image of a quadratic polynomial…

Let $d\geq 3$. The two-dimensional inverse sieve theorem considers subsets of $\{0,\ldots,N\}^{2}$ occupying at most $\alpha p$ residue classes modulo every prime, and concludes th…

Let $S\subseteq \{0,\ldots,N\}$ satisfy $|S|\geq N^{\varepsilon}$ and occupy fewer than $\alpha p$ residue classes modulo every prime $p$, where $0<\alpha<1$. Helfgott–Venkatesh an…