arXiv subjects
Publications and source records attributed to Andrew Lott.
For positive integers $r, m$ and $N$, every $r$-coloring of $\{1, \dots, N\}$ contains a monochromatic solution to $x_1+\dots+x_{m+1}=y_1+\dots+y_m$ provided that $N \ge 3^r (r!)^{1/m}$, which is qualitatively optimal when $m$ is logarithmic in $r$.
Let $S_N=\{1\le d\le N:d=x^2+y^2\text{ for some }x,y\in\mathbb Z\}.$ We prove a power-saving form of the van der Corput property for $S_N$. As a consequence, we obtain a strong S\'{a}rk\"{o}zy-type result: if $A\subseteq [N]$ has no nonzero difference equal to a sum of two squares, then $|A|\ll_\varepsilon N^{7/8+\varepsilon}$ for every $\varepsilon>0$, improving upon an earlier quasipolynomial bound due to Rice. The shape of this bound is optimal, as a construction of Younis yields a set $A\subseteq [N]$ with $|A|\gg N^{1/2}$ such that $(A-A)\cap S_N=\emptyset$.
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 \[ A_n g(x):=\frac{1}{q^n}\sum_{\substack{f\in \mathbb{F}_q[t]\\\deg f 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: \[ \big\|\sup_{n\in\mathbb{N}}|A_ng|\big\|_{L^2(X)} \leq C_2\|g\|_{L^2(X)} \qquad \left( g\in L^2(X)\right), \] 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 L\^e-Liu-Wooley and Champagne-Ge-L\^e-Liu-Wooley. These refinements are of independent interest.
Green and Sawhney recently obtained a quasipolynomial bound in the Furstenberg--S\'ark\"ozy theorem for square differences by proving an ``arithmetic level-d'' inequality, thereby yielding a greatly improved density increment scheme. We apply their method to treat general intersective polynomials $h\in\mathbb{Z}[x]$. In particular, let \[ D(h(\mathbb{N}),X):= \max{|A|:\ A\subseteq [1,X]\cap\mathbb{N} \text{and}\ (A-A)\cap h(\mathbb{N})\subseteq\{0\}}. \] We prove that for every $0<\mu<1/2$ there are constants $c_0, X_{\text{min}}>0$ depending on $h$ and $\mu$ such that for every $X>X_{\text{min}}$, \[D(h(\mathbb{N}), X)\leq Xe^{-c_0(\log X)^\mu}.\] This is the best quantitative upper bound presently known for sets lacking intersective polynomial differences, improving upon the work of Arala. In order to achieve the admissible exponent range $0<\mu<1/2$, we use sieve methods to develop novel exponential sum estimates in the style of Rice, and we use the ``random sparsification'' procedure of Green and Sawhney.
Fix a positive prime power $q$, and let $\mathbb{F}_q[t]$ be the ring of polynomials over the finite field $\mathbb{F}_q$ with $\text{char}(\mathbb{F}_q)>2$. Suppose $A \subseteq \{f \in \mathbb{F}_q[t]: \text{deg } f \leq N\}$ contains no pair of elements whose difference is of the form $P-1$ with $P$ irreducible. Adapting Green's approach to S\'ark\"ozy'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 L\^{e} and Spencer. An important distinction between Green's argument and ours lies in the properties of exponential sums over function fields, which differ in several interesting ways from their number-field counterparts.
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 \textit{intersective} if $D(X,N)=o(N)$, with the most well-studied examples being $X=S=\{n^2: n\in \mathbb{N}\}$ and $X=\mathcal{P}-1=\{p-1: p\text{ prime}\}$. In addition to some new results, including exact formulas and estimates for $D(X,N)$ in some non-intersective cases like $X=\mathcal{P}$ and $X=S+k$, $k\in \mathbb{N}$, we also provide a comprehensive survey of known bounds and extensive computational data. In particular, we utilize an existing algorithm for finding maximum cliques in graphs to determine $D(S,N)$ for $N\leq 300$ and $D(\mathcal{P}-1,N)$ for $N\leq 500$. None of these exact values appear previously in the literature.
By Maynard's theorem and the subsequent improvements by the Polymath Project, there exists a positive integer $b\leq 246$ such that there are infinitely many primes $p$ such that $p+b$ is also prime. Let $P_1,...,P_t\in \mathbb{Z}[y]$ 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 \leq 246 $ such that $x+P_1(y),x+P_2(y),...,x+P_t(y)$ and $x+P_1(y)+b,x+P_2(y)+b,...,x+P_t(y)+b$ are all prime. Our work is inspired by Pintz, who proved a similar result for the special case of arithmetic progressions.
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 - the d-fold direct product of the primes - of positive relative upper density. We show that A contains all polynomial configurations of the form $x+P_0(y)v_0,\ldots, x+P_l(y)v_l$, 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.
For $A\subseteq \mathbb{R}$, let $A+A=\{a+b: a,b\in A\}$ and $AA=\{ab: a,b\in A\}$. For $k\in \mathbb{N}$, let $SP(k)$ denote the minimum value of $\max\{|A+A|, |AA|\}$ over all $A\subseteq \mathbb{N}$ with $|A|=k$. Here we establish $SP(k)=3k-3$ for $2\leq k \leq 7$, the $k=7$ case achieved for example by $\{1,2,3,4,6,8,12\}$, while $SP(k)=3k-2$ for $k=8,9$, the $k=9$ case achieved for example by $\{1,2,3,4,6,8,9,12,16\}$. For $4\leq k \leq 7$, we provide two proofs using different applications of Freiman's $3k-4$ theorem; one of the proofs 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 $r>1$, with separate treatments of the overlapping cases $r\neq 2$ and $r\geq 2$.
A $\textit{covering system}$ is a collection of integer congruences such that every integer satisfies at least one congruence in the collection. A covering system is called $\textit{distinct}$ if all of its moduli are distinct. An expansive literature has developed on covering systems since their introduction by Erd\H{o}s. 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 $11$.
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 $k\geq d+1$, 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 $r\in\{2,3,4\}$.
For integers $k,r\geq 2$, 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 on a monochromatic subgraph on $k$ vertices. Here we make a careful effort of extracting explicit upper bounds for $R_r(k)$ from the pigeonhole principle alone. Our main term improves on previously documented explicit bounds for $r\geq 3$, 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 $R_r(k)\leq \left(\frac{3+e}{2}\right)\frac{(r(k-2))!}{((k-2)!)^r}(1+o_{r\to \infty}(1)),$ and we conclude by noting that our methods combine with previous estimates on $R_r(3)$ to improve the constant $\frac{3+e}{2}$ to $\frac{3+e}{2}-\frac{d}{48}$, where $d=66-R_4(3)\geq 4$. We also compare our formulas, and previously documented formulas, to some collected numerical data.