11 problems

Let $S$ be the set of squares, and let $S_t(Q/t)$ be the subset defined earlier in the paper. For $M\in\mathbb{Z}$, $N\in\mathbb{N}$, and a complex sequence $(a_n)$, one has … wher…

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 $\Xi_4(A,B)$ denote the quartic large sieve quantity for parameters $A,B\geq 1$, and let $\varepsilon>0$. Quartic large sieve upper-bound conjecture. For any $A,B\geq 1$, … The…

Weighted dual quadratic large sieve conjecture. For every $\varepsilon>0$,

Let $\boldsymbol{F}_q[t]$ be the polynomial ring over the finite field with $q$ elements. Let $\boldsymbol{S}$ be a set of nonzero monic polynomials in $\boldsymbol{F}_q[t]$ of deg…

Let $A\subset [1,N]$ be a set of integers. For each prime $p$, write $A_p$ for the set of residue classes modulo $p$ represented by elements of $A$. The notation $X\ll Y$ means tha…

Refined inverse conjecture for the large sieve. Under these assumptions,

Green's inverse conjecture for the large sieve. Then $|A|\ll X^\epsilon$ unless $A$ is contained, apart from a set of size $X^\epsilon$, in the set of values of a quadratic polynom…

Let $\mathscr{A}$ be a set of positive integers satisfying … for all sufficiently large primes $p$. Symmetric inverse large sieve conjecture. Either there is a rational quadratic…

Let $X_0\in\mathbb{N}$, let $\rho>0$, and let $X\in\mathbb{N}$ be sufficiently large in terms of $X_0$ and $\rho$. Suppose that $A,B\subset[X]$ satisfy … for all…