Canadian Journal of Mathematics最新文献

{"title":"On restricted Falconer distance sets","authors":"José Gaitan, Allan Greenleaf, Eyvindur Ari Palsson, Georgios Psaromiligkos","doi":"10.4153/s0008414x24000117","DOIUrl":"https://doi.org/10.4153/s0008414x24000117","url":null,"abstract":"<p>We introduce a class of Falconer distance problems, which we call of restricted type, lying between the classical version and its pinned variant. Prototypical restricted distance sets are the diagonal distance sets, <span>k</span>-point configuration sets given by <span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226123035304-0552:S0008414X24000117:S0008414X24000117_eqnu1.png\"><span data-mathjax-type=\"texmath\"><span>$$ begin{align*}Delta^{mathrm{diag}}(E)= { ,|(x,x,dots,x)-(y_1,y_2,dots,y_{k-1})| : x, y_1, dots,y_{k-1} in E, }end{align*} $$</span></span></img></span>for a compact <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226123035304-0552:S0008414X24000117:S0008414X24000117_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$Esubset mathbb {R}^d$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226123035304-0552:S0008414X24000117:S0008414X24000117_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$kge 3$</span></span></img></span></span>. We show that <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226123035304-0552:S0008414X24000117:S0008414X24000117_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$Delta ^{mathrm{diag}}(E)$</span></span></img></span></span> has non-empty interior if the Hausdorff dimension of <span>E</span> satisfies <span><span>(0.1)</span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226123035304-0552:S0008414X24000117:S0008414X24000117_eqn1.png\"><span data-mathjax-type=\"texmath\"><span>$$ begin{align} dim(E)&gt; begin{cases} frac{2d+1}3, &amp; k=3, frac{(k-1)d}k,&amp; kge 4. end{cases} end{align} $$</span></span></img></span>We prove an extension of this to <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226123035304-0552:S0008414X24000117:S0008414X24000117_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$C^omega $</span></span></img></span></span> Riemannian metrics <span>g</span> close to the product of Euclidean metrics. For product metrics, this follows from known results on pinned distance sets, but to obtain a result for general perturbations <span>g</span>, we present a sequence of proofs of partial results, leading up to the proof of the full result, which is based on estimates for multilinear Fourier integral operators.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"81 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140001427","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Tracial oscillation zero and stable rank one","authors":"Xuanlong Fu, Huaxin Lin","doi":"10.4153/s0008414x24000099","DOIUrl":"https://doi.org/10.4153/s0008414x24000099","url":null,"abstract":"<p>Let <span>A</span> be a separable (not necessarily unital) simple <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240221010039813-0084:S0008414X24000099:S0008414X24000099_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$C^*$</span></span></img></span></span>-algebra with strict comparison. We show that if <span>A</span> has tracial approximate oscillation zero, then <span>A</span> has stable rank one and the canonical map <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240221010039813-0084:S0008414X24000099:S0008414X24000099_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$Gamma $</span></span></img></span></span> from the Cuntz semigroup of <span>A</span> to the corresponding lower-semicontinuous affine function space is surjective. The converse also holds. As a by-product, we find that a separable simple <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240221010039813-0084:S0008414X24000099:S0008414X24000099_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$C^*$</span></span></img></span></span>-algebra which has almost stable rank one must have stable rank one, provided it has strict comparison and the canonical map <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240221010039813-0084:S0008414X24000099:S0008414X24000099_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$Gamma $</span></span></img></span></span> is surjective.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"190 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139918029","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the cross-product conjecture for the number of linear extensions","authors":"Swee Hong Chan, Igor Pak, Greta Panova","doi":"10.4153/s0008414x24000087","DOIUrl":"https://doi.org/10.4153/s0008414x24000087","url":null,"abstract":"<p>We prove a weak version of the <span>cross-product conjecture</span>: <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240209164228069-0556:S0008414X24000087:S0008414X24000087_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$textrm {F}(k+1,ell ) hskip .06cm textrm {F}(k,ell +1) ge (frac 12+varepsilon ) hskip .06cm textrm {F}(k,ell ) hskip .06cm textrm {F}(k+1,ell +1)$</span></span></img></span></span>, where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240209164228069-0556:S0008414X24000087:S0008414X24000087_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$textrm {F}(k,ell )$</span></span></img></span></span> is the number of linear extensions for which the values at fixed elements <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240209164228069-0556:S0008414X24000087:S0008414X24000087_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$x,y,z$</span></span></img></span></span> are <span>k</span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240209164228069-0556:S0008414X24000087:S0008414X24000087_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$ell $</span></span></img></span></span> apart, respectively, and where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240209164228069-0556:S0008414X24000087:S0008414X24000087_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$varepsilon&gt;0$</span></span></img></span></span> depends on the poset. We also prove the converse inequality and disprove the <span>generalized cross-product conjecture</span>. The proofs use geometric inequalities for mixed volumes and combinatorics of words.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139771607","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Heights and quantitative arithmetic on stacky curves","authors":"Brett Nasserden, Stanley Yao Xiao","doi":"10.4153/s0008414x24000075","DOIUrl":"https://doi.org/10.4153/s0008414x24000075","url":null,"abstract":"<p>In this paper, we investigate the theory of heights in a family of stacky curves following recent work of Ellenberg, Satriano, and Zureick-Brown. We first give an elementary construction of a height which is seen to be dual to theirs. We count rational points having bounded ESZ-B height on a particular stacky curve, answering a question of Ellenberg, Satriano, and Zureick-Brown. We also show that when the Euler characteristic of stacky curves is non-positive, the ESZ-B height coming from the anti-canonical divisor class fails to have the Northcott property. We prove that a stacky version of a conjecture of Vojta is equivalent to the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240219131415699-0390:S0008414X24000075:S0008414X24000075_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$abc$</span></span></img></span></span>-conjecture.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"54 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139918194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On definable groups and D-groups in certain fields with a generic derivation","authors":"Ya’acov Peterzil, Anand Pillay, Françoise Point","doi":"10.4153/s0008414x24000063","DOIUrl":"https://doi.org/10.4153/s0008414x24000063","url":null,"abstract":"<p>We continue our study from Peterzil et al. (2022, <span>Preprint</span>, arXiv:2208.08293) of finite-dimensional definable groups in models of the theory <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240202124840300-0805:S0008414X24000063:S0008414X24000063_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$T_{partial }$</span></span></img></span></span>, the model companion of an o-minimal <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240202124840300-0805:S0008414X24000063:S0008414X24000063_inline2.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal {L}}$</span></span></img></span></span>-theory <span>T</span> expanded by a generic derivation <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240202124840300-0805:S0008414X24000063:S0008414X24000063_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$partial $</span></span></img></span></span> as in Fornasiero and Kaplan (2021, <span>Journal of Mathematical Logic</span> 21, 2150007).</p><p>We generalize Buium’s notion of an algebraic <span>D</span>-group to <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240202124840300-0805:S0008414X24000063:S0008414X24000063_inline4.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal {L}}$</span></span></img></span></span>-definable <span>D</span>-groups, namely <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240202124840300-0805:S0008414X24000063:S0008414X24000063_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$(G,s)$</span></span></img></span></span>, where <span>G</span> is an <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240202124840300-0805:S0008414X24000063:S0008414X24000063_inline6.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal {L}}$</span></span></img></span></span>-definable group in a model of <span>T</span>, and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240202124840300-0805:S0008414X24000063:S0008414X24000063_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$s:Gto tau (G)$</span></span></img></span></span> is an <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240202124840300-0805:S0008414X24000063:S0008414X24000063_inline8.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal {L}}$</span></span></img></span></span>-definable group section. Our main theorem says that every definable group of finite dimension in a model of <span>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139688708","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"FALTINGS’ MAIN -ADIC COMPARISON THEOREMS FOR NON-SMOOTH SCHEMES","authors":"Tongmu He","doi":"10.4153/s0008414x24000051","DOIUrl":"https://doi.org/10.4153/s0008414x24000051","url":null,"abstract":"","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"24 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139531795","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Refinements of Katz–Sarnak theory for the number of points on curves over finite fields","authors":"Jonas Bergström, Everett W. Howe, Elisa Lorenzo García, Christophe Ritzenthaler","doi":"10.4153/s0008414x2400004x","DOIUrl":"https://doi.org/10.4153/s0008414x2400004x","url":null,"abstract":"<p>This paper goes beyond Katz–Sarnak theory on the distribution of curves over finite fields according to their number of rational points, theoretically, experimentally, and conjecturally. In particular, we give a formula for the limits of the moments measuring the asymmetry of this distribution for (non-hyperelliptic) curves of genus <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226123836538-0613:S0008414X2400004X:S0008414X2400004X_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$ggeq 3$</span></span></img></span></span>. The experiments point to a stronger notion of convergence than the one provided by the Katz–Sarnak framework for all curves of genus <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226123836538-0613:S0008414X2400004X:S0008414X2400004X_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$geq 3$</span></span></img></span></span>. However, for elliptic curves and for hyperelliptic curves of every genus, we prove that this stronger convergence cannot occur.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"51 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140001425","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Trigonometric convexity of the multidimensional indicator","authors":"Aleksandr Mkrtchyan, Armen Vagharshakyan","doi":"10.4153/s0008414x24000014","DOIUrl":"https://doi.org/10.4153/s0008414x24000014","url":null,"abstract":"<p>The notion of indicator of an analytic function, that describes the function’s growth along rays, was introduced by Phragmen and Lindelöf. Trigonometric convexity is a defining property of the indicator. For multivariate cases, an analogous property of trigonometric convexity was not known so far. We prove the property of trigonometric convexity for the indicator of multivariate analytic functions, introduced by Ivanov. The results that we obtain are sharp. Derivation of a multidimensional analogue of the inverse Fourier transform in a sector and obtaining estimates on its decay is an important step of our proof.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139552494","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Systems involving mean value formulas on trees","authors":"Alfredo Miranda, Carolina A. Mosquera, Julio D. Rossi","doi":"10.4153/s0008414x23000913","DOIUrl":"https://doi.org/10.4153/s0008414x23000913","url":null,"abstract":"<p>In this paper, we study the Dirichlet problem for systems of mean value equations on a regular tree. We deal both with the directed case (the equations verified by the components of the system at a node in the tree only involve values of the unknowns at the successors of the node in the tree) and the undirected case (now the equations also involve the predecessor in the tree). We find necessary and sufficient conditions on the coefficients in order to have existence and uniqueness of solutions for continuous boundary data. In a particular case, we also include an interpretation of such solutions as a limit of value functions of suitable two-players zero-sum games.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"82 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139552733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Meridional rank and bridge number of knotted 2-spheres","authors":"Jason Joseph, Puttipong Pongtanapaisan","doi":"10.4153/s0008414x23000883","DOIUrl":"https://doi.org/10.4153/s0008414x23000883","url":null,"abstract":"<p>The meridional rank conjecture asks whether the bridge number of a knot in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240123135603811-0282:S0008414X23000883:S0008414X23000883_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$S^3$</span></span></img></span></span> is equal to the minimal number of meridians needed to generate the fundamental group of its complement. In this paper, we investigate the analogous conjecture for knotted spheres in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240123135603811-0282:S0008414X23000883:S0008414X23000883_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$S^4$</span></span></img></span></span>. Towards this end, we give a construction to produce classical knots with quotients sending meridians to elements of any finite order in Coxeter groups and alternating groups, which detect their meridional ranks. We establish the equality of bridge number and meridional rank for these knots and knotted spheres obtained from them by twist-spinning. On the other hand, we show that the meridional rank of knotted spheres is not additive under connected sum, so that either bridge number also collapses, or meridional rank is not equal to bridge number for knotted spheres.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"60 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139552492","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}