17 problems

Zhao's conjecture. The right-hand side should be replaceable by

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

Large-sieve bound conjecture. One has

Let $S(x)=\sum_{n\leq N}a_n e(nx)$, let $Z=\sum_{n\leq N}|a_n|^2$, and let ${\cal S}$ be a set of primes at most $Q$ with cardinality $S$. Elliott's conjecture. The bound … should…

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 $\Xi_4(A,B)$ 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 $A\subset [N]$ be a Sidon set, and for each prime $p$ let $A_p$ denote the set of residue classes modulo $p$ occupied by $A$. Suppose that $|A_p|\leq \alpha p$ for every prime…

Let $k$ be the degree parameter, let $Q,N$ be positive parameters, and let $\Delta_{k,1}(Q,N)$ denote the one-dimensional large-sieve constant for polynomial moduli. Aistleitner–Ay…

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 $\mathcal{N}$ be a set of $Z$ integers contained in $[M+1,M+N]$, and let $\omega(p)$ denote the number of residue classes modulo $p$ containing no element of $\mathcal{N}$. Def…

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…

Let $A$ be the large sieve matrix, and let $A^{\star}A$ be the associated $N$ by $N$ positive definite matrix. Suppose that $N\sim \alpha Q^2$ as $Q\to\infty$ for some finite posit…

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…