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Joel Moreira

Publications and source records attributed to Joel Moreira.

33 records · Page 2Linked to original sources
arXiv2021

A decomposition of multicorrelation sequences for commuting transformations along primes

We study multicorrelation sequences arising from systems with commuting transformations. Our main result is a refinement of a decomposition result of Frantzikinakis and it states that any multicorrelation sequences for commuting transformations can be decomposed, for every $ε>0$, as the sum of a nilsequence $ϕ(n)$ and a sequence $ω(n)$ satisfying $\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N |ω(n)|<ε$ and $\lim_{N\to\infty}\frac{1}{|\mathbb{P}\cap [N]|}\sum_{p\in \mathbb{P}\cap [N]} |ω(p)|<ε$.

math.DS↗
arXiv2021

A spectral refinement of the Bergelson-Host-Kra decomposition and new multiple ergodic theorems

We investigate how spectral properties of a measure preserving system $(X,\mathcal{B},μ,T)$ are reflected in the multiple ergodic averages arising from that system. For certain sequences $a:\mathbb{N}\to\mathbb{N}$ we provide natural conditions on the spectrum $σ(T)$ such that for all $f_1,\ldots,f_k\in L^\infty$, \begin{equation*} \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N \prod_{j=1}^k T^{ja(n)}f_j = \lim_{N\rightarrow\infty} \frac{1}{N} \sum_{n=1}^N \prod_{j=1}^k T^{jn}f_j \end{equation*} in $L^2$-norm. In particular, our results apply to infinite arithmetic progressions $a(n)=qn+r$, Beatty sequences $a(n)=\lfloor θn+γ\rfloor$, the sequence of squarefree numbers $a(n)=q_n$, and the sequence of prime numbers $a(n)=p_n$. We also obtain a new refinement of Szemerédi's theorem via Furstenberg's correspondence principle. ERRATUM: Theorem 7.1 in the paper is incorrect as stated, and the error originates with Proposition 7.5, part (iii), which was incorrectly quoted from the literature. In the attached erratum we fix the problem by establishing a slightly weaker version of Theorem 7.1 and use it to give a new proof of Theorem 4.2. This ensures that all main results in our main article remain correct. We thank Zhengxing Lian and Jiahao Qiu for bringing this mistake to our attention.

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arXiv2021

On Rado conditions for nonlinear Diophantine equations

Building on previous work of Di Nasso and Luperi Baglini, we provide general necessary conditions for a Diophantine equation to be partition regular. These conditions are inspired by Rado's characterization of partition regular linear homogeneous equations. We conjecture that these conditions are also sufficient for partition regularity, at least for equations whose corresponding monovariate polynomial is linear. This would provide a natural generalization of Rado's theorem. We verify that such a conjecture hold for the equations $x^{2}-xy+ax+by+cz=0$ and $x^{2}-y^{2}+ax+by+cz=0$ for $a,b,c\in \mathbb{Z}$ such that $abc=0$ or $% a+b+c=0$. To deal with these equations, we establish new results concerning the partition regularity of polynomial configurations in $\mathbb{Z}$ such as $\left\{ x,x+y,xy+x+y\right\} $, building on the recent result on the partition regularity of $\left\{ x,x+y,xy\right\} $.

math.CO↗
arXiv2020

Structure of multicorrelation sequences with integer part polynomial iterates along primes

Let $T$ be a measure preserving $\mathbb{Z}^\ell$-action on the probability space $(X,{\mathcal B},μ),$ $q_1,\dots,q_m:{\mathbb R}\to{\mathbb R}^\ell$ vector polynomials, and $f_0,\dots,f_m\in L^\infty(X)$. For any $ε> 0$ and multicorrelation sequences of the form $\displaystyleα(n)=\int_Xf_0\cdot T^{ \lfloor q_1(n) \rfloor }f_1\cdots T^{ \lfloor q_m(n) \rfloor }f_m\;dμ$ we show that there exists a nilsequence $ψ$ for which $\displaystyle\lim_{N - M \to \infty} \frac{1}{N-M} \sum_{n=M}^{N-1} |α(n) - ψ(n)| \leq ε$ and $\displaystyle\lim_{N \to \infty} \frac{1}{π(N)} \sum_{p \in {\mathbb P}\cap[1,N]} |α(p) - ψ(p)| \leq ε.$ This result simultaneously generalizes previous results of Frantzikinakis [2] and the authors [11,13].

math.DS↗
arXiv2020

Single and multiple recurrence along non-polynomial sequences

We establish new recurrence and multiple recurrence results for a rather large family $\mathcal{F}$ of non-polynomial functions which includes tempered functions defined in [11], as well as functions from a Hardy field with the property that for some $\ell\in \mathbb{N}\cup\{0\}$, $\lim_{x\to\infty }f^{(\ell)}(x)=\pm\infty$ and $\lim_{x\to\infty }f^{(\ell+1)}(x)=0$. Among other things, we show that for any $f\in\mathcal{F}$, any invertible probability measure preserving system $(X,\mathcal{B},μ,T)$, any $A\in\mathcal{B}$ with $μ(A)>0$, and any $ε>0$, the sets of returns $$ R_{ε, A}= \big\{n\in\mathbb{N}:μ(A\cap T^{-\lfloor f(n)\rfloor}A)>μ^2(A)-ε\big\} $$ and $$ R^{(k)}_{A}= \big\{ n\in\mathbb{N}: μ\big(A\cap T^{\lfloor f(n)\rfloor}A\cap T^{\lfloor f(n+1)\rfloor}A\cap\cdots\cap T^{\lfloor f(n+k)\rfloor}A\big)>0\big\} $$ possess somewhat unexpected properties of largeness; in particular, they are thick, i.e., contain arbitrarily long intervals.

math.DS↗
arXiv2019

Optimal lower bounds for multiple recurrence

Let $(X, \mathcal{B},μ,T)$ be an ergodic measure preserving system, $A \in \mathcal{B}$ and $ε>0$. We study the largeness of sets of the form \begin{equation*} \begin{split} S = \left\{ n\in\mathbb{N}\colonμ(A\cap T^{-f_1(n)}A\cap T^{-f_2(n)}A\cap\ldots\cap T^{-f_k(n)}A)> μ(A)^{k+1} - ε\right\} \end{split} \end{equation*} for various families $\{f_1,\dots,f_k\}$ of sequences $f_i\colon \mathbb{N} \to \mathbb{N}$. For $k \leq 3$ and $f_{i}(n)=i f(n)$, we show that $S$ has positive density if $f(n)=q(p_n)$ where $q \in \mathbb{Z}[x]$ satisfies $q(1)$ or $q(-1) =0$ and $p_n$ denotes the $n$-th prime; or when $f$ is a certain Hardy field sequence. If $T^q$ is ergodic for some $q \in \mathbb{N}$, then for all $r \in \mathbb{Z}$, $S$ is syndetic if $f(n) = qn + r$. For $f_{i}(n)=a_{i}n$, where $a_{i}$ are distinct integers, we show that $S$ can be empty for $k\geq 4$, and for $k = 3$ we found an interesting relation between the largeness of $S$ and the abundance of solutions to certain linear equations in sparse sets of integers. We also provide some partial results when the $f_{i}$ are distinct polynomials.

math.DS↗
arXiv2019

A proof of a sumset conjecture of Erdős

In this paper we show that every set $A \subset \mathbb{N}$ with positive density contains $B+C$ for some pair $B,C$ of infinite subsets of $\mathbb{N}$, settling a conjecture of Erdős. The proof features two different decompositions of an arbitrary bounded sequence into a structured component and a pseudo-random component. Our methods are quite general, allowing us to prove a version of this conjecture for countable amenable groups.

math.CO↗
arXiv2016

New polynomial and multidimensional extensions of classical partition results

In the 1970s Deuber introduced the notion of $(m,p,c)$-sets in $\mathbb{N}$ and showed that these sets are partition regular and contain all linear partition regular configurations in $\mathbb{N}$. In this paper we obtain enhancements and extensions of classical results on $(m,p,c)$-sets in two directions. First, we show, with the help of ultrafilter techniques, that Deuber's results extend to polynomial configurations in abelian groups. In particular, we obtain new partition regular polynomial configurations in $\mathbb{Z}^d$. Second, we give two proofs of a generalization of Deuber's results to general commutative semigroups. We also obtain a polynomial version of the central sets theorem of Furstenberg, extend the theory of $(m,p,c)$-systems of Deuber, Hindman and Lefmann and generalize a classical theorem of Rado regarding partition regularity of linear systems of equations over $\mathbb{N}$ to commutative semigroups.

math.CO↗
arXiv2016

Monochromatic sums and products in $\mathbb{N}$

An old question in Ramsey theory asks whether any finite coloring of the natural numbers admits a monochromatic pair $\{x+y,xy\}$. We answer this question affirmatively in a strong sense by exhibiting a large new class of non-linear patterns which can be found in a single cell of any finite partition of $\mathbb{N}$. Our proof involves a correspondence principle which transfers the problem into the language of topological dynamics. As a corollary of our main theorem we obtain partition regularity for new types of equations, such as $x^2-y^2=z$ and $x^2+2y^2-3z^2=w$.

math.CO↗
arXiv2016

Large subsets of discrete hypersurfaces in $\mathbb{Z}^d$ contain arbitrarily many collinear points

In 1977 L.T. Ramsey showed that any sequence in $\mathbb{Z}^2$ with bounded gaps contains arbitrarily many collinear points. Thereafter, in 1980, C. Pomerance provided a density version of this result, relaxing the condition on the sequence from having bounded gaps to having gaps bounded on average. We give a higher dimensional generalization of these results. Our main theorem is the following. Theorem: Let $d\in\mathbb{N}$, let $f:\mathbb{Z}^d\to\mathbb{Z}^{d+1}$ be a Lipschitz map and let $A\subset\mathbb{Z}^d$ have positive upper Banach density. Then $f(A)$ contains arbitrarily many collinear points. Note that Pomerance's theorem corresponds to the special case $d=1$. In our proof, we transfer the problem from a discrete to a continuous setting, allowing us to take advantage of analytic and measure theoretic tools such as Rademacher's theorem.

math.CO↗
arXiv2015

Measure preserving actions of affine semigroups and {x+y,xy} patterns

Ergodic and combinatorial results obtained in [10] involved measure preserving actions of the affine group ${\mathcal A}_K$ of a countable field $K$. In this paper we develop a new approach based on ultrafilter limits which allows one to refine and extend the results obtained in [10] to a more general situation involving the measure preserving actions of the non-amenable affine semigroups of a large class of integral domains. (The results in [10] heavily depend on the amenability of the affine group of a field). Among other things, we obtain, as a corollary of an ultrafilter ergodic theorem, the following result: Let $K$ be a number field and let ${\mathcal O}_K$ be the ring of integers of $K$. For any finite partition $K=C_1\cup\cdots\cup C_r$ there exists $i\in\{1,\dots,r\}$ and many $x\in K$ and $y\in{\mathcal O}_K$ such that $\{x+y,xy\}\subset C_i$.

math.DS↗
arXiv2015

Ergodic Theorem involving additive and multiplicative groups of a field and $\{x+y,xy\}$ patterns

We establish a "diagonal" ergodic theorem involving the additive and multiplicative groups of a countable field $K$ and, with the help of a new variant of Furstenberg's correspondence principle, prove that any "large" set in $K$ contains many configurations of the form $\{x+y,xy\}$. We also show that for any finite coloring of $K$ there are many $x,y\in K$ such that $x,x+y$ and $xy$ have the same color. Finally, by utilizing a finitistic version of our main ergodic theorem, we obtain combinatorial results pertaining to finite fields. In particular we obtain an alternative proof for a result obtained by Cilleruelo [11], showing that for any finite field $F$ and any subsets $E_1,E_2\subset F$ with $|E_1||E_2|>6|F|$, there exist $u,v\in F$ such that $u+v\in E_1$ and $uv\in E_2$.

math.CO↗
arXiv2014

A conditional construction of restricted isometries

We study the restricted isometry property of a matrix that is built from the discrete Fourier transform matrix by collecting rows indexed by quadratic residues. We find an $ε>0$ such that, conditioned on a folklore conjecture in number theory, this matrix satisfies the restricted isometry property with sparsity parameter $K=Ω(M^{1/2+ε})$, where $M$ is the number of rows.

math.FA↗
arXiv2014

Derandomizing restricted isometries via the Legendre symbol

The restricted isometry property (RIP) is an important matrix condition in compressed sensing, but the best matrix constructions to date use randomness. This paper leverages pseudorandom properties of the Legendre symbol to reduce the number of random bits in an RIP matrix with Bernoulli entries. In this regard, the Legendre symbol is not special---our main result naturally generalizes to any small-bias sample space. We also conjecture that no random bits are necessary for our Legendre symbol--based construction.

math.CO↗