Large subsets of discrete hypersurfaces in Zd\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.Comment: 16 pages, small part of the argument clarified in light of suggestions from the refere

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 $(X,\mathcal{B},\mu,T)$, any $A\in\mathcal{B}$ with $\mu(A)>0$, and any $\epsilon>0$, the sets of returns $$R_{\epsilon, A}= \big\{n\in\mathbb{N}:\mu(A\cap T^{-\lfloor f(n)\rfloor}A)>\mu^2(A)-\epsilon\big\}$$ and $$R^{(k)}_{A}= \big\{ n\in\mathbb{N}: \mu\big(A\cap T^{\lfloor f(n)\rfloor}A\cap T^{\lfloor f(n+1)\rfloor}A\cap\cdots\cap T^{\lfloor f(n+k)\rfloor}A\big)>0\big\}$$ possess somewhat unexpected properties of largeness; in particular, they are thick, i.e., contain arbitrarily long intervals.Comment: 51 page

Disjointness for measurably distal group actions and applications

We generalize Berg's notion of quasi-disjointness to actions of countable groups and prove that every measurably distal system is quasi-disjoint from every measure preserving system. As a corollary we obtain easy to check necessary and sufficient conditions for two systems to be disjoint, provided one of them is measurably distal. We also obtain a Wiener--Wintner type theorem for countable amenable groups with distal weights and applications to weighted multiple ergodic averages and multiple recurrence.Comment: 28 page

RNA Transport (Partly) Revealed!

AbstractSpecific mRNAs are transported to dendrites where their translation may modify synaptic plasticity. In this issue of Neuron, Kanai et al. use affinity chromatography and mass spectrometry to identify a large number of new factors that associate with kinesin, a molecular motor, and employ siRNA technology to demonstrate their importance for RNA transport in neurons

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 $(X,{\mathcal B},\mu),$ $q_1,\dots,q_m:{\mathbb R}\to{\mathbb R}^\ell$ vector polynomials, and $f_0,\dots,f_m\in L^\infty(X)$. For any $\epsilon > 0$ and multicorrelation sequences of the form $\displaystyle\alpha(n)=\int_Xf_0\cdot T^{ \lfloor q_1(n) \rfloor }f_1\cdots T^{ \lfloor q_m(n) \rfloor }f_m\;d\mu$ we show that there exists a nilsequence $\psi$ for which $\displaystyle\lim_{N - M \to \infty} \frac{1}{N-M} \sum_{n=M}^{N-1} |\alpha(n) - \psi(n)| \leq \epsilon$ and $\displaystyle\lim_{N \to \infty} \frac{1}{\pi(N)} \sum_{p \in {\mathbb P}\cap[1,N]} |\alpha(p) - \psi(p)| \leq \epsilon.$ This result simultaneously generalizes previous results of Frantzikinakis [2] and the authors [11,13].Comment: 7 page

Pausing on Polyribosomes: Make Way for Elongation in Translational Control

Among the three phases of mRNA translation—initiation, elongation, and termination—initiation has traditionally been considered to be rate limiting and thus the focus of regulation. Emerging evidence, however, demonstrates that control of ribosome translocation (polypeptide elongation) can also be regulatory and indeed exerts a profound influence on development, neurologic disease, and cell stress. The correspondence of mRNA codon usage and the relative abundance of their cognate tRNAs is equally important for mediating the rate of polypeptide elongation. Here, we discuss recent results showing that ribosome pausing is a widely used mechanism for controlling translation and, as a result, biological transitions in health and disease

CPEB controls oocyte growth and follicle development in the mouse

CPEB is a sequence-specific RNA-binding protein that regulates polyadenylation-induced translation. In Cpeb knockout mice, meiotic progression is disrupted at pachytene due to inhibited translation of synaptonemal complex protein mRNAs. To assess the function of CPEB after pachytene, we used the zona pellucida 3 (Zp3) promoter to generate transgenic mice expressing siRNA that induce the destruction of Cpeb mRNA. Oocytes from these animals do not develop normally; they undergo parthenogenetic cell division in the ovary, exhibit abnormal polar bodies, are detached from the cumulus granulosa cell layer, and display spindle and nuclear anomalies. In addition, many follicles contain apoptotic granulosa cells. CPEB binds several oocyte mRNAs, including Smad1, Smad5, spindlin, Bub1b, Mos, H1foo, Obox1, Dnmt1o, TiParp, Trim61 and Gdf9, a well described oocyte-expressed growth factor that is necessary for follicle development. In Cpeb knockdown oocytes, Gdf9 RNA has a shortened poly(A) tail and reduced expression. These data indicate that CPEB controls the expression of Gdf9 mRNA, which in turn is necessary for oocyte-follicle development. Finally, several phenotypes, i.e. progressive oocyte loss and infertility, elicited by the knockdown of CPEB in oocytes resemble those of the human premature ovarian failure syndrome

The Mos pathway regulates cytoplasmic polyadenylation in Xenopus oocytes

Cytoplasmic polyadenylation controls the translation of several maternal mRNAs during Xenopus oocyte maturation and requires two sequences in the 3\u27 untranslated region (UTR), the U-rich cytoplasmic polyadenylation element (CPE), and the hexanucleotide AAUAAA. c-mos mRNA is polyadenylated and translated soon after the induction of maturation, and this protein kinase is necessary for a kinase cascade culminating in cdc2 kinase (MPF) activation. Other mRNAs are polyadenylated later, around the time of cdc2 kinase activation. To determine whether there is a hierarchy in the cytoplasmic polyadenylation of maternal mRNAs, we ablated c-mos mRNA with an antisense oligonucleotide. This prevented histone B4 and cyclin A1 and B1 mRNA polyadenylation, indicating that the polyadenylation of these mRNAs is Mos dependent. To investigate a possible role of cdc2 kinase in this process, cyclin B was injected into oocytes lacking c-mos mRNA. cdc2 kinase was activated, but mitogen-activated protein kinase was not. However, polyadenylation of cyclin B1 and histone B4 mRNA was still observed. This demonstrates that cdc2 kinase can induce cytoplasmic polyadenylation in the absence of Mos. Our data further indicate that although phosphorylation of the CPE binding protein may be involved in the induction of Mos-dependent polyadenylation, it is not required for Mos-independent polyadenylation. We characterized the elements conferring Mos dependence (Mos response elements) in the histone B4 and cyclin B1 mRNAs by mutational analysis. For histone B4 mRNA, the Mos response elements were in the coding region or 5\u27 UTR. For cyclin B1 mRNA, the main Mos response element was a CPE that overlaps with the AAUAAA hexanucleotide. This indicates that the position of the CPE can have a profound influence on the timing of cytoplasmic polyadenylation

A proof of a sumset conjecture of Erd\H{o}s

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 Erd\H{o}s. 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.Comment: 54 pages. Corrected proof of Theorem 3.22 and added Example 3.27 Keywords: sum sets, almost periodic functions, ultrafilter