Research

We prove a function-field analogue of Bourgain's $L^2$ pointwise ergodic theorem. Let $q$ be a power of a prime $p$, let $\mathbb{F}_q[t]$ be the ring of polynomials over the finite field $\mathbb{F}_q$, and let $\mathbb{F}_q[t][u]$ be the ring of polynomials over $\mathbb{F}_q[t]$. Let $T^{(1)},\ldots,T^{(\ell)}$ be commuting, measure-preserving $\mathbb{F}_q[t]$-actions on a $\sigma$-finite measure space $(X,\mu)$, and let $P_1,\ldots,P_\ell\in \mathbb{F}_q[t][u]\setminus\{0\}$.

Define a sequence of operators $(A_n)_{n\in \mathbb{N}}$ by

We prove that $(A_n)_{n\in\mathbb{N}}$ satisfies an $L^2$ oscillation ergodic theorem:

where the constant $C_1>0$ depends only on $P_1,\ldots,P_\ell$ and $q$. This in particular implies that the sequence $(A_ng(x))_{n\in\mathbb{N}}$ converges for almost every $x\in X$ and that $(A_n)_{n\in\mathbb{N}}$ satisfies an $L^2$ maximal inequality:

where the constant $C_2>0$ depends only on $P_1,\ldots,P_\ell$ and $q$. Our tools include the circle method in function fields and refinements of Weyl sum estimates in this setting, further developing the work of Lê-Liu-Wooley and Champagne-Ge-Lê-Liu-Wooley. These refinements are of independent interest.

Very recently, Green and Sawhney obtained a quasipolynomial bound in the Furstenberg--Sárközy theorem for square differences by proving an ''arithmetic level-$d$'' inequality, thereby yielding a greatly improved density increment scheme. We adapt their method to general intersective polynomials $h\in\mathbb{Z}[x]$ and obtain an analogous quasipolynomial upper bound for the largest subset of $\{1,2,\dots,X\}$ whose difference set contains no nonzero element of the form $h(n)$ with $n\in \mathbb{Z}$. This is the best quantitative upper bound presently known for sets lacking intersective polynomial differences. In contrast to the square case, extending the method to general intersective polynomials requires performing a density increment iteration in which the underlying polynomial changes at each step; a key contribution of this paper is to show that the arithmetic level-$d$ inequality remains effective uniformly across all auxiliary polynomials arising in the iteration. We also develop smoothly weighted versions of the exponential sum estimates of Rice.

Fix a positive prime power $q$, and let $\mathbb{F}_q[t]$ be the ring of polynomials over the finite field $\mathbb{F}_q$. Suppose $A \subseteq \{f \in \mathbb{F}_q[t]\colon\deg f \le N\}$ contains no pair of elements whose difference is of the form $P-1$ with $P$ irreducible. Adapting Green's approach to Sárközy's theorem for shifted primes in $\mathbb{Z}$ using the van der Corput property, we show that \[|A| \ll q^{(N+1)(11/12+o(1))},\] improving upon the bound $O\big(q^{(1-c/\log N)(N+1)}\big)$ due to Lê and Spencer.

We provide a multidimensional extension of previous results on the existence of polynomial progressions in dense subsets of the primes. Let $A$ be a subset of the prime lattice $\mathbb{P}^d$ of positive relative upper density. We show that $A$ contains all polynomial configurations of the form $x + P_0(y)v_0, \ldots, x + P_\ell(y)v_\ell$, for some $x \in \mathbb{Z}^d$ and $y \in \mathbb{N}$, which satisfy a certain non-degeneracy condition. We also obtain quantitative bounds on the size of such polynomial configuration, if $A$ is a subset of the first $N$ positive integers.

By Maynard's theorem and the subsequent improvements by the Polymath Project, there exists a positive integer $b \le 246$ such that there are infinitely many primes $p$ such that $p + b$ is also prime. Let $P_1, \ldots, P_t \in \mathbb{Z}[y]$ with $P_1(0) = \cdots = P_t(0) = 0$. We use the transference argument of Tao and Ziegler to prove there exist positive integers $x$, $y$, and $b \le 246$ such that $x + P_1(y)$, $x + P_2(y)$, $\ldots$, $x + P_t(y)$ and $(x + P_1(y)) + b$, $\ldots$, $(x + P_t(y)) + b$ are all prime. Our work is inspired by Pintz, who proved a similar result for the special case of arithmetic progressions.

Abstract. We explore from several perspectives the following question: given $X\subseteq \mathbb{Z}$ and $N\in \mathbb{N}$, what is the maximum size $D(X,N)$ of $A\subseteq \{1,2,\dots,N\}$ before $A$ is forced to contain two distinct elements that differ by an element of $X$? The set of forbidden differences, $X$, is called intersective if $D(X,N)=o(N)$, with the most well-studied examples being $X=S=\{n^2: n\in \mathbb{N}\}$ and $X=\mathcal{P}-1=\{p-1: p\text{ prime}\}$. We study $D(X,N)$ and the structure of extremal sets when $X$ is a finite set, a union of arithmetic progressions, or a translate of a multiplicative set, and give formulas and estimates in a number of cases. We also describe several computational explorations that illuminate the range of possible behaviors.

Abstract. The sum–product problem involves finding the minimum of $\max\{|A+A|,|AA|\}$ for finite sets $A\subset\mathbb{R}$. For $k=|A|$, define $\mathrm{SP}(k)$ to be this minimum over all such $A$ with $|A|=k$. We show $\mathrm{SP}(k)=3k-3$ for $2\le k\le 7$ and $\mathrm{SP}(k)=3k-2$ for $k\in\{8,9\}$. For $4\le k\le7$, we give two proofs using different applications of Freiman's $3k-4$ theorem; one proof includes extensive case analysis on the product sets of $k$-element subsets of $(2k-3)$-term arithmetic progressions. For $k=8,9$, we apply Freiman's $3k-3$ theorem for product sets, and investigate the sumset of the union of two geometric progressions with the same common ratio $r>1$, treating separately the overlapping cases $r\ne2$ and $r\ge2$.

Abstract. For integers $k,r\ge 2$, the diagonal Ramsey number $R_r(k)$ is the minimum $N\in\mathbb{N}$ such that every $r$-coloring of the edges of a complete graph on $N$ vertices yields a monochromatic subgraph on $k$ vertices. Here we make a careful effort to extract explicit upper bounds for $R_r(k)$ from the pigeonhole principle alone. Our main term improves on previously documented explicit bounds for $r\ge3$, and we also consider an often-ignored secondary term, which allows us to subtract a uniformly bounded-below positive proportion of the main term. Asymptotically, we give a self-contained proof that $$R_r(k)\le \Big(\tfrac{3+e}{2}\Big)\,\frac{(r(k-2))!}{((k-2)!)^r}\,(1+o_{r\to\infty}(1)),$$ and we note our methods combine with previous estimates on $R_r(3)$ to improve the constant $\tfrac{3+e}{2}$ to $\tfrac{3+e}{2}-\tfrac{d}{48}$, where $d=66-R_4(3)\ge 4$. We also compare our formulas, and previously documented formulas, to numerical data.

Abstract. A covering system is a collection of integer congruences such that every integer satisfies at least one congruence in the collection. A covering system is called distinct if all of its moduli are distinct. An expansive literature has developed on covering systems since their introduction by Erdős. Here we provide a full classification of distinct covering systems with at most ten moduli, which we group together based on two forms of equivalence. As a consequence, we determine the minimum cardinality of a distinct covering system with all moduli exceeding $2$, which is $11$.

Abstract. A standard proof of Schur’s Theorem yields that any $r$-coloring of $\{1,2,\dots,R_r-1\}$ yields a monochromatic solution to $x+y=z$, where $R_r$ is the classical $r$-color Ramsey number, the minimum $N$ such that any $r$-coloring of a complete graph on $N$ vertices yields a monochromatic triangle. We explore generalizations and modifications of this result in higher dimensional integer lattices, showing in particular that if $k\ge d+1$, then any $r$-coloring of $\{1,2,\dots,R_r(k)^d-1\}^d$ yields a monochromatic solution to $x_1+\cdots+x_{k-1}=x_k$ with $\{x_1,\dots,x_d\}$ linearly independent, where $R_r(k)$ is the analogous Ramsey number in which triangles are replaced by complete graphs on $k$ vertices. We also obtain computational results and examples in the case $d=2$, $k=3$, and $r\in\{2,3,4\}$.