Bourgain in his seminal paper of 1986 about the analysis of maximal functions associated to convex bodies has estimated in a sharp way the $L^{2}$-operator norm of the maximal function associated to a kernel $Kin L^{1},$ with differentiable Fourier transform $widehat{K}.$ We formulate the extension to Bourgain’s $L^{2}$-estimate in the setting of maximal functions on graded Lie groups. Our criterion is formulated in terms of the group Fourier transform of the kernel. We discuss the application of our main result to the $L^{p}$-boundedness of maximal functions on graded Lie groups.
{"title":"L 2-Maximal Functions on Graded Lie Groups","authors":"Duván Cardona","doi":"10.1093/imrn/rnae105","DOIUrl":"https://doi.org/10.1093/imrn/rnae105","url":null,"abstract":"Bourgain in his seminal paper of 1986 about the analysis of maximal functions associated to convex bodies has estimated in a sharp way the $L^{2}$-operator norm of the maximal function associated to a kernel $Kin L^{1},$ with differentiable Fourier transform $widehat{K}.$ We formulate the extension to Bourgain’s $L^{2}$-estimate in the setting of maximal functions on graded Lie groups. Our criterion is formulated in terms of the group Fourier transform of the kernel. We discuss the application of our main result to the $L^{p}$-boundedness of maximal functions on graded Lie groups.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153000","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $F$ be a free group of rank $r$ and fix some $win F$. For any compact group $G$ we can define a measure $mu _{w,G}$ on $G$ by (Haar-)uniformly sampling $g_{1},...,g_{r}in G$ and evaluating $w(g_{1},...,g_{r})$. In [23], Magee and Puder study the behavior of the moments of $mu _{w,U(n)}$ as a function of $n$, establishing a connection between their asymptotic behavior and certain algebraic invariants of $w$, such as its commutator length. We employ geometric insights to refine their analysis, and show that the asymptotic behavior of the moments is also governed by the primitivity rank of $w$. Additionally, we also apply our methods to prove a special case of a conjecture of Hanany and Puder [13, Conjecture 1.13] regarding the asymptotic behavior of expected values of irreducible characters of $U(n)$ under $mu _{w,U(n)}$.
{"title":"Word Measures on Unitary Groups: Improved Bounds for Small Representations","authors":"Yaron Brodsky","doi":"10.1093/imrn/rnae100","DOIUrl":"https://doi.org/10.1093/imrn/rnae100","url":null,"abstract":"Let $F$ be a free group of rank $r$ and fix some $win F$. For any compact group $G$ we can define a measure $mu _{w,G}$ on $G$ by (Haar-)uniformly sampling $g_{1},...,g_{r}in G$ and evaluating $w(g_{1},...,g_{r})$. In [23], Magee and Puder study the behavior of the moments of $mu _{w,U(n)}$ as a function of $n$, establishing a connection between their asymptotic behavior and certain algebraic invariants of $w$, such as its commutator length. We employ geometric insights to refine their analysis, and show that the asymptotic behavior of the moments is also governed by the primitivity rank of $w$. Additionally, we also apply our methods to prove a special case of a conjecture of Hanany and Puder [13, Conjecture 1.13] regarding the asymptotic behavior of expected values of irreducible characters of $U(n)$ under $mu _{w,U(n)}$.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141152988","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $gamma (t)=(P_{1}(t),ldots ,P_{n}(t))$ where $P_{i}$ is a real polynomial with zero constant term for each $1leq ileq n$. We will show the existence of the configuration ${x,x+gamma (t)}$ in sets of positive density $epsilon $ in $[0,1]^{n}$ with a gap estimate $tgeq delta (epsilon )$ when $P_{i}$’s are arbitrary, and in $[0,N]^{n}$ with a gap estimate $tgeq delta (epsilon )N^{n}$ when $P_{i}$’s are of distinct degrees where $delta (epsilon )=exp left (-exp left (cepsilon ^{-4}right )right )$ and $c$ only depends on $gamma $. To prove these two results, decay estimates of certain oscillatory integral operators and Bourgain’s reduction are primarily utilised. For the first result, dimension-reducing arguments are also required to handle the linear dependency. For the second one, we will prove a stronger result instead, since then an anisotropic rescaling is allowed in the proof to eliminate the dependence of the decay estimate on $N$. And as a byproduct, using the strategy token to prove the latter case, we extend the corner-type Roth theorem previously proven by the first author and Guo.
{"title":"Two-Point Polynomial Patterns in Subsets of Positive Density in $mathbb{R}^{n}$","authors":"Xuezhi Chen, Changxing Miao","doi":"10.1093/imrn/rnae108","DOIUrl":"https://doi.org/10.1093/imrn/rnae108","url":null,"abstract":"\u0000 Let $gamma (t)=(P_{1}(t),ldots ,P_{n}(t))$ where $P_{i}$ is a real polynomial with zero constant term for each $1leq ileq n$. We will show the existence of the configuration ${x,x+gamma (t)}$ in sets of positive density $epsilon $ in $[0,1]^{n}$ with a gap estimate $tgeq delta (epsilon )$ when $P_{i}$’s are arbitrary, and in $[0,N]^{n}$ with a gap estimate $tgeq delta (epsilon )N^{n}$ when $P_{i}$’s are of distinct degrees where $delta (epsilon )=exp left (-exp left (cepsilon ^{-4}right )right )$ and $c$ only depends on $gamma $. To prove these two results, decay estimates of certain oscillatory integral operators and Bourgain’s reduction are primarily utilised. For the first result, dimension-reducing arguments are also required to handle the linear dependency. For the second one, we will prove a stronger result instead, since then an anisotropic rescaling is allowed in the proof to eliminate the dependence of the decay estimate on $N$. And as a byproduct, using the strategy token to prove the latter case, we extend the corner-type Roth theorem previously proven by the first author and Guo.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141115728","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recently, motivated by the theory of real local Arthur packets, making use of the wavefront sets of representations over non-Archimedean local fields $F$, Ciubotaru, Mason-Brown, and Okada defined the weak local Arthur packets consisting of certain unipotent representations and conjectured that they are unions of local Arthur packets. In this paper, we prove this conjecture for $textrm{Sp}_{2n}(F)$ and split $textrm{SO}_{2n+1}(F)$ with the assumption of the residue field characteristic of $F$ being large. In particular, this implies the unitarity of these unipotent representations. We also discuss the generalization of the weak local Arthur packets beyond unipotent representations, which reveals the close connection with a conjecture of Jiang on the structure of wavefront sets for representations in local Arthur packets.
{"title":"On the Weak Local Arthur Packets Conjecture for Split Classical Groups","authors":"Baiying Liu, Chi-Heng Lo","doi":"10.1093/imrn/rnae098","DOIUrl":"https://doi.org/10.1093/imrn/rnae098","url":null,"abstract":"\u0000 Recently, motivated by the theory of real local Arthur packets, making use of the wavefront sets of representations over non-Archimedean local fields $F$, Ciubotaru, Mason-Brown, and Okada defined the weak local Arthur packets consisting of certain unipotent representations and conjectured that they are unions of local Arthur packets. In this paper, we prove this conjecture for $textrm{Sp}_{2n}(F)$ and split $textrm{SO}_{2n+1}(F)$ with the assumption of the residue field characteristic of $F$ being large. In particular, this implies the unitarity of these unipotent representations. We also discuss the generalization of the weak local Arthur packets beyond unipotent representations, which reveals the close connection with a conjecture of Jiang on the structure of wavefront sets for representations in local Arthur packets.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141126640","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $(M^{n},g,f)$ be a Ricci shrinker such that $text{Ric}_{f}=frac{1}{2}g$ and the measure induced by the weighted volume element $(4pi )^{-frac{n}{2}}e^{-f}dv_{g}$ is a probability measure. Given a point $pin M$, we consider two probability measures defined in the tangent space $T_{p}M$, namely the Gaussian measure $gamma $ and the measure $overline{nu }$ induced by the exponential map of $M$ to $p$. In this paper, we prove a result that provides an upper estimate for the Wasserstein distance with respect to the Euclidean metric $g_{0}$ between the measures $overline{nu }$ and $gamma $, and which also elucidates the rigidity implications resulting from this estimate.
{"title":"The Wasserstein Distance for Ricci Shrinkers","authors":"Franciele Conrado, Detang Zhou","doi":"10.1093/imrn/rnae099","DOIUrl":"https://doi.org/10.1093/imrn/rnae099","url":null,"abstract":"Let $(M^{n},g,f)$ be a Ricci shrinker such that $text{Ric}_{f}=frac{1}{2}g$ and the measure induced by the weighted volume element $(4pi )^{-frac{n}{2}}e^{-f}dv_{g}$ is a probability measure. Given a point $pin M$, we consider two probability measures defined in the tangent space $T_{p}M$, namely the Gaussian measure $gamma $ and the measure $overline{nu }$ induced by the exponential map of $M$ to $p$. In this paper, we prove a result that provides an upper estimate for the Wasserstein distance with respect to the Euclidean metric $g_{0}$ between the measures $overline{nu }$ and $gamma $, and which also elucidates the rigidity implications resulting from this estimate.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141059552","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Galen Dorpalen-Barry, Nicholas Proudfoot, Jidong Wang
We give a cohomological interpretation of the Heaviside filtration on the Varchenko–Gelfand ring of a pair $({mathcal{A}},{mathcal{K}})$, where ${mathcal{A}}$ is a real hyperplane arrangement and ${mathcal{K}}$ is a convex open subset of the ambient vector space. This builds on work of the first author, who studied the filtration from a purely algebraic perspective, as well as work of Moseley, who gave a cohomological interpretation in the special case where ${mathcal{K}}$ is the ambient vector space. We also define the Gelfand–Rybnikov ring of a conditional oriented matroid, which simultaneously generalizes the Gelfand–Rybnikov ring of an oriented matroid and the aforementioned Varchenko–Gelfand ring of a pair. We give purely combinatorial presentations of the ring, its associated graded, and its Rees algebra.
{"title":"Equivariant Cohomology and Conditional Oriented Matroids","authors":"Galen Dorpalen-Barry, Nicholas Proudfoot, Jidong Wang","doi":"10.1093/imrn/rnad025","DOIUrl":"https://doi.org/10.1093/imrn/rnad025","url":null,"abstract":"We give a cohomological interpretation of the Heaviside filtration on the Varchenko–Gelfand ring of a pair $({mathcal{A}},{mathcal{K}})$, where ${mathcal{A}}$ is a real hyperplane arrangement and ${mathcal{K}}$ is a convex open subset of the ambient vector space. This builds on work of the first author, who studied the filtration from a purely algebraic perspective, as well as work of Moseley, who gave a cohomological interpretation in the special case where ${mathcal{K}}$ is the ambient vector space. We also define the Gelfand–Rybnikov ring of a conditional oriented matroid, which simultaneously generalizes the Gelfand–Rybnikov ring of an oriented matroid and the aforementioned Varchenko–Gelfand ring of a pair. We give purely combinatorial presentations of the ring, its associated graded, and its Rees algebra.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141059806","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study an interacting system of competing particles on the real line. Two populations of positive and negative particles evolve according to branching Brownian motion. When opposing particles meet, their charges neutralize and the particles annihilate, as in an inert chemical reaction. We show that, with positive probability, the two populations coexist and that, on this event, the interface is asymptotically linear with a random slope. A variety of generalizations and open problems are discussed.
{"title":"Annihilating Branching Brownian Motion","authors":"Daniel Ahlberg, Omer Angel, Brett Kolesnik","doi":"10.1093/imrn/rnae068","DOIUrl":"https://doi.org/10.1093/imrn/rnae068","url":null,"abstract":"We study an interacting system of competing particles on the real line. Two populations of positive and negative particles evolve according to branching Brownian motion. When opposing particles meet, their charges neutralize and the particles annihilate, as in an inert chemical reaction. We show that, with positive probability, the two populations coexist and that, on this event, the interface is asymptotically linear with a random slope. A variety of generalizations and open problems are discussed.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937362","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Correction to: The Stability of Relativistic Fluids in Linearly Expanding Cosmologies","authors":"","doi":"10.1093/imrn/rnae102","DOIUrl":"https://doi.org/10.1093/imrn/rnae102","url":null,"abstract":"","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141000942","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Matija Bucić, Tung Nguyen, Alex Scott, Paul Seymour
In 1977, Erd̋s and Hajnal made the conjecture that, for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ has a clique or stable set of size at least $|G|^{c}$, and they proved that this is true with $ |G|^{c}$ replaced by $2^{csqrt{log |G|}}$. Until now, there has been no improvement on this result (for general $H$). We prove a strengthening: that for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ with $|G|ge 2$ has a clique or stable set of size at least $$ begin{align*} &2^{csqrt{log |G|loglog|G|}}.end{align*} $$ Indeed, we prove the corresponding strengthening of a theorem of Fox and Sudakov, which in turn was a common strengthening of theorems of Rödl, Nikiforov, and the theorem of Erd̋s and Hajnal mentioned above.
{"title":"Induced Subgraph Density. I. A loglog Step Towards Erd̋s–Hajnal","authors":"Matija Bucić, Tung Nguyen, Alex Scott, Paul Seymour","doi":"10.1093/imrn/rnae065","DOIUrl":"https://doi.org/10.1093/imrn/rnae065","url":null,"abstract":"In 1977, Erd̋s and Hajnal made the conjecture that, for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ has a clique or stable set of size at least $|G|^{c}$, and they proved that this is true with $ |G|^{c}$ replaced by $2^{csqrt{log |G|}}$. Until now, there has been no improvement on this result (for general $H$). We prove a strengthening: that for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ with $|G|ge 2$ has a clique or stable set of size at least $$ begin{align*} &2^{csqrt{log |G|loglog|G|}}.end{align*} $$ Indeed, we prove the corresponding strengthening of a theorem of Fox and Sudakov, which in turn was a common strengthening of theorems of Rödl, Nikiforov, and the theorem of Erd̋s and Hajnal mentioned above.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937447","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct the Arthur packets for symplectic and even orthogonal similitude groups over a $p$-adic field and show that they are stable and satisfy the twisted endoscopic character relations.
{"title":"Arthur Packets for Quasisplit GSp(2n) and GO(2n) Over a p-Adic Field","authors":"Bin Xu","doi":"10.1093/imrn/rnae086","DOIUrl":"https://doi.org/10.1093/imrn/rnae086","url":null,"abstract":"We construct the Arthur packets for symplectic and even orthogonal similitude groups over a $p$-adic field and show that they are stable and satisfy the twisted endoscopic character relations.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937450","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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