arXiv subjects
Publications and source records attributed to Florian K. Richter.
We show that the set $\mathbb{P}-1$ of shifted primes contains infinitely many sum-product patterns of the form $\{x,x+y,xy\}$ with $x,y$ arbitrarily large distinct integers. More strongly, we can also show that, for any $k\geq 1$, the set $\mathbb{P}-1$ contains longer patterns of the form $\{x,x+y,\ldots, x+ky,xy\}$ with $x,y$ arbitrarily large distinct integers, a statement that contains the Green--Tao theorem as a special case.
Fix $s\in\mathbb{N}$. We prove that there exists a subadditive density on $\mathbb{N}$ such that, for every polynomial $P\in\mathbb{Z}[y]$ with $P(0)=0$, every set of positive density contains configurations $\{x,x+P(y),xy^s\}$ for arbitrarily large $x>y\geq 2$. When $s=1$, this strengthens Moreira's partition-regularity result for $\{x,x+P(y),xy\}$ to a density theorem, and when $s>1$ this yields new partition-regularity results.
Let $Ω(n)$ denote the number of prime factors of a positive integer $n$ counted with multiplicities. We show that for any bounded functions $a,b\colon\mathbb{N}\to\mathbb{C}$, $$\frac{1}{\log{N}}\sum_{n=1}^N \frac{a(Ω(n))b(Ω(n+1))}{n} = \Bigg(\frac{1}{N}\sum_{n=1}^N a(Ω(n))\Bigg)\Bigg(\frac{1}{N}\sum_{n=1}^N b(Ω(n))\Bigg) + \mathrm{o}_{N\to\infty}(1).$$ This generalizes a theorem of Tao on the logarithmically averaged two-point correlation Chowla conjecture. Our result is quantitative and the explicit error term that we obtain establishes double-logarithmic savings. As an application, we obtain new results about the distribution of $Ω(p+1)$ as $p$ ranges over $\ell$-almost primes for a "typical" value of $\ell$.
We give a short proof of the fact that every set of natural numbers with positive upper Banach density contains the sum of two infinite sets. The approach simplifies earlier existing proofs.
For a wide range of functions $W\colon\mathbb{N}\to\mathbb{N}$, we establish a general result for estimating weighted averages of the form\[\mathbb{E}^{W}_{n \le N} f(\vartheta(n))= \frac{1}{W(N)}\sum_{n=1}^N (W(n)-W(n-1))f(\vartheta(n)),\]where $f\colon \{1,\ldots,N\}\to\mathbb{C}$ is an arbitrary function, and $\vartheta(n)$ is any arithmetic function that adheres to a certain Gaussian distribution condition. (For instance, one may take $\vartheta(n)=Ω(n)$, where $Ω(n)$ counts the number of prime factors of $n$ with multiplicity, or $\vartheta(n)=s_q(p_n)$, where $s_q$ is the sum-of-digits function in base $q$ and $p_n$ denotes the $n$-th prime. Additional natural examples are discussed in the paper.) Building on our main theorem, we show that if $h(n)$ is a function from a Hardy field with polynomial growth then $(h(\vartheta(n)))_{n\in\mathbb{N}}$ is uniformly distributed mod $1$ if and only if one of the following (mutually exclusive) conditions is satisfied: (i) $\lim_{x\to\infty} \frac{|h(x)-p(x)|}{x \log x}=\infty$ for all $p(x)\in \mathbb{Q}[x]$; (ii) $\lim_{x\to\infty}\frac{|h(x)-p(x)|}{\sqrt{x}}=\infty$ for each $p(x)\in \mathbb{Q}[x]$ and there exists $q(x)\in \mathbb{Q}[x]$ such that $\lim_{x\to\infty}\frac{|h(x)-q(x)|}{x}<\infty$. This leads to several novel applications. For example, it follows that $(Ω(n)^c)_{n\in\mathbb{N}}$ is uniformly distributed mod $1$ if and only if $c$ is a non-integer greater than $\frac{1}{2}$.
We investigate the structure of return-time sets determined by orbits along polynomial tuples in minimal topological dynamical systems. Building on the topological characteristic factor theory of Glasner, Huang, Shao, Weiss, and Ye, we prove a structure theorem showing that, in a minimal system, return-time sets coincide -- up to a non-piecewise syndetic set -- with those in its maximal infinite-step pronilfactor. As applications, we establish three new multiple recurrence theorems concerning linear recurrence along dynamically defined syndetic sets and polynomial recurrence along arithmetic progressions in minimal and totally minimal systems. We also show how our main theorem can be used to prove that two previously separate conjectures -- one due to Glasner, Huang, Shao, Weiss, and Ye and the other due to Leibman -- are equivalent.
Let $d(\cdot)$ denote the natural density on the positive integers. We characterize all sets $A,B$ with positive density satisfying $d(A+B)=d(A)+d(B)$, under the assumption that the two sets are not both contained in a proper finite union of residue classes. This gives a new inverse theorem for Kneser's sumset inequality in the integers, and provides a partial answer to a long-standing open question of Erdős and Graham.
Alice and Bob are given $n$-bit integer pairs $(x,y)$ and $(a,b)$, respectively, and they must decide if $y=ax+b$. We prove that the randomised communication complexity of this Point--Line Incidence problem is $Θ(\log n)$. This confirms a conjecture of Cheung, Hatami, Hosseini, and Shirley (CCC 2023) that the complexity is super-constant, and gives the first example of a communication problem with constant support-rank but super-constant randomised complexity.
Van der Waerden's theorem, published in Nieuw. Arch. Wisk. 15 (1927), acted as a catalyst for major further developments in Ramsey theory. In this survey, we delve into the legacy of this mathematical gem, tracing its historical origin, exploring its wide-ranging connections across research areas, and highlighting some of the important results it has inspired.
We obtain new results pertaining to convergence and recurrence of multiple ergodic averages along functions from a Hardy field. Among other things, we confirm some of the conjectures posed by Frantzikinakis in [Fra10; Fra16] and obtain combinatorial applications which contain, as rather special cases, several previously known (polynomial and non-polynomial) extensions of Szemeredi's theorem on arithmetic progressions [BL96; BLL08; FW09; Fra10; BMR17]. One of the novel features of our results, which is not present in previous work, is that they allow for a mixture of polynomials and non-polynomial functions. As an illustration, assume $f_i(t)=a_{i,1}t^{c_{i,1}}+\cdots+a_{i,d}t^{c_{i,d}}$ for $c_{i,j}>0$ and $a_{i,j}\in\mathbb{R}$. Then $\bullet$ for any measure preserving system $(X,\mathcal{B},μ,T)$ and $h_1,\dots,h_k\in L^\infty(X)$, the limit $$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N T^{[f_1(n)]}h_1\cdots T^{[f_k(n)]}h_k$$ exists in $L^2$; $\bullet$ for any $E\subset \mathbb{N}$ with $\overline{\mathrm{d}}(E)>0$ there are $a,n\in\mathbb{N}$ such that $\{a,\, a+[f_1(n)],\ldots,a+[f_k(n)]\}\subset E$. We also show that if $f_1,\dots,f_k$ belong to a Hardy field, have polynomial growth, and are such that no linear combination of them is a polynomial, then for any measure preserving system $(X,{\mathcal B},μ,T)$ and any $A\in{\mathcal B}$, $$\limsup_{N\to\infty}\frac{1}{N}\sum_{n=1}^Nμ\Big(A\cap T^{-[ f_1(n) ]}A\cap\ldots\cap T^{-[f_k(n)]}A\Big)\,\geq\,μ(A)^{k+1}.$$
We develop an analytic approach that draws on tools from Fourier analysis and ergodic theory to study Ramsey-type problems involving sums and products in the integers. Suppose $Q$ denotes a polynomial with integer coefficients. We establish two main results. First, we show that if $Q(1) = 0$, then any set of natural numbers with positive upper logarithmic density contains a pair of the form $\{x + Q(y), xy\}$ for some $x, y \in \mathbb{N} \setminus \{1\}$. Second, we prove that if $Q(0) = 0$, then any set of natural numbers with positive density relative to a new multiplicative notion of density, which arises naturally in the context of such problems, contains $\{x + Q(y), xy\}$ for some $x, y \in \mathbb{N}$.
For any set $A$ of natural numbers with positive upper Banach density and any $k\geq 1$, we show the existence of an infinite set $B\subset{\mathbb N}$ and a shift $t\geq0$ such that $A-t$ contains all sums of $m$ distinct elements from $B$ for all $m\in\{1,\ldots,k\}$. This can be viewed as a density analog of Hindman's finite sums theorem. Our proof reveals the natural relationships among infinite sumsets, the dynamics underpinning arithmetic progressions, and homogeneous spaces of nilpotent Lie groups.
In contrast to finite arithmetic configurations, relatively little is known about which infinite patterns can be found in every set of natural numbers with positive density. Building on recent advances showing infinite sumsets can be found, we explore numerous open problems and obstructions to finding other infinite configurations in every set of natural numbers with positive density.
By juxtaposing ideas from fractal geometry and dynamical systems, Furstenberg proposed a series of conjectures in the late 1960's that explore the relationship between digit expansions with respect to multiplicatively independent bases. In this work, we introduce and study - in the discrete context of the integers - analogues of some of the notions and results surrounding Furstenberg's work. In particular, we define a new class of fractal sets of integers that parallels the notion of $\times r$-invariant sets on the 1-torus and investigate the additive and geometric independence between two such fractal sets when they are structured with respect to multiplicatively independent bases. Our main results in this direction parallel the works of Furstenberg, Hochman-Shmerkin, Shmerkin, Wu, and Lindenstrauss-Meiri-Peres and include: -a classification of all subsets of the positive integers that are simultaneously $\times r$- and $\times s$-invariant; -integer analogues of two of Furstenberg's transversality conjectures pertaining to the dimensions of the intersection $A\cap B$ and the sumset $A+B$ of $\times r$- and $\times s$-invariant sets $A$ and $B$ when $r$ and $s$ are multiplicatively independent; and -a description of the dimension of iterated sumsets $A+A+\cdots+A$ for any $\times r$-invariant set $A$. We achieve these results by combining ideas from fractal geometry and ergodic theory to build a bridge between the continuous and discrete regimes. For the transversality results, we rely heavily on quantitative bounds on the $L^q$-dimensions of projections of restricted digit Cantor measures obtained recently by Shmerkin. We end by outlining a number of open questions and directions regarding fractal subsets of the integers.
Originating in harmonic analysis, interpolation sets were first studied in dynamics by Glasner and Weiss in the 1980s. A set $S \subset \mathbb{N}$ is an interpolation set for a class of topological dynamical systems $\mathcal{C}$ if any bounded sequence on $S$ can be extended to a sequence that arises from a system in $\mathcal{C}$. In this paper, we provide combinatorial characterizations of interpolation sets for: $\bullet$ (totally) minimal systems; $\bullet$ topologically (weak) mixing systems; $\bullet$ strictly ergodic systems; and $\bullet$ zero entropy systems. Additionally, we prove some results on a slightly different notion, called weak interpolation sets, for several classes of systems. We also answer a question of Host, Kra, and Maass concerning the connection between sets of pointwise recurrence for distal systems and $IP$-sets.
Motivated by questions asked by Erdos, we prove that any set $A\subset{\mathbb N}$ with positive upper density contains, for any $k\in{\mathbb N}$, a sumset $B_1+\cdots+B_k$, where $B_1,\dots,B_k\subset{\mathbb N}$ are infinite. Our proof uses ergodic theory and relies on structural results for measure preserving systems. Our techniques are new, even for the previously known case of $k=2$.
We give a new proof of a sumset conjecture of Furstenberg that was first proved by Hochman and Shmerkin in 2012: if $\log r / \log s$ is irrational and $X$ and $Y$ are $\times r$- and $\times s$-invariant subsets of $[0,1]$, respectively, then $\dim_\text{H} (X+Y) = \min ( 1, \dim_\text{H} X + \dim_\text{H} Y)$. Our main result yields information on the size of the sumset $λX + ηY$ uniformly across a compact set of parameters at fixed scales. The proof is combinatorial and avoids the machinery of local entropy averages and CP-processes, relying instead on a quantitative, discrete Marstrand projection theorem and a subtree regularity theorem that may be of independent interest.
We study equidistribution properties of translations on nilmanifolds along functions of polynomial growth from a Hardy field. More precisely, if $X=G/Γ$ is a nilmanifold, $a_1,\ldots,a_k\in G$ are commuting nilrotations, and $f_1,\ldots,f_k$ are functions of polynomial growth from a Hardy field then we show that $\bullet$ the distribution of the sequence $a_1^{f_1(n)}\cdot\ldots\cdot a_k^{f_k(n)}Γ$ is governed by its projection onto the maximal factor torus, which extends Leibman's Equidistribution Criterion form polynomials to a much wider range of functions; and $\bullet$ the orbit closure of $a_1^{f_1(n)}\cdot\ldots\cdot a_k^{f_k(n)}Γ$ is always a finite union of sub-nilmanifolds, which extends some of the previous work of Leibman and Frantzikinakis on this topic.