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11
Counting lattice points
, 2006
"... Abstract. For a locally compact second countable group G and a lattice subgroup Γ, we give an explicit quantitative solution of the lattice point counting problem in general domains in G, provided that i) G has finite upper local dimension, and the domains satisfy a basic regularity condition, ii) t ..."
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Abstract. For a locally compact second countable group G and a lattice subgroup Γ, we give an explicit quantitative solution of the lattice point counting problem in general domains in G, provided that i) G has finite upper local dimension, and the domains satisfy a basic regularity condition, ii) the mean ergodic theorem for the action of G on G/Γ holds, with a rate of convergence. The error term we establish matches the best current result for balls in symmetric spaces of simple higherrank Lie groups, but holds in much greater generality. A significant advantage of the ergodic theoretic approach we use is that the solution to the lattice point counting problem is uniform over families of lattice subgroups provided they admit a uniform spectral gap. In particular, the uniformity property holds for families of finite index subgroups satisfying a quantitative variant of property τ. We discuss a number of applications, including: counting lattice points in general domains in semisimple Salgebraic groups, counting rational points on group varieties with respect to a height function, and quantitative angular (or conical) equidistribution of lattice points in symmetric spaces and in affine symmetric varieties. We note that the mean ergodic theorems which we establish are based on spectral methods, including the spectral transfer principle and the KunzeStein phenomenon. We formulate and prove appropriate analogues of both of these results in the setup of adele groups, and they constitute a necessary step in our proof of quantitative results in counting rational points. Contents
Type III1 equilibrium states of the Toeplitz algebra of the affine semigroup over the natural numbers
 J. Funct. Anal
"... Abstract. We complete the analysis of KMSstates of the Toeplitz algebra T (N o N×) of the affine semigroup over the natural numbers, recently studied by Raeburn and the first author, by showing that for every inverse temperature β in the critical interval 1 ≤ β ≤ 2, the unique KMSβstate is of type ..."
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Cited by 11 (3 self)
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Abstract. We complete the analysis of KMSstates of the Toeplitz algebra T (N o N×) of the affine semigroup over the natural numbers, recently studied by Raeburn and the first author, by showing that for every inverse temperature β in the critical interval 1 ≤ β ≤ 2, the unique KMSβstate is of type III1. We prove this by reducing the type classification from T (N o N×) to that of the symmetric part of the BostConnes system, with a shift in inverse temperature. To carry out this reduction we first obtain a parametrization of the Nica spectrum of N o N × in terms of an adelic space. Combining a characterization of traces on crossed products due to the second author with an analysis of the action of N o N × on the Nica spectrum, we can also recover all the KMSstates of T (N o N×) originally computed by Raeburn and the first author. Our computation sheds light on why there is a free transitive circle action on the extremal KMSβstates for β> 2 that does not ostensibly come from an action of T on the C∗algebra.
Phase transition in the ConnesMarcolli GL2system
"... Abstract. We develop a general framework for analyzing KMSstates on C ∗algebras arising from actions of Hecke pairs. We then specialize to the system recently introduced by Connes and Marcolli and classify its KMSstates for inverse temperatures β ̸ = 0,1. In particular, we show that for each β ∈ ..."
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Cited by 10 (5 self)
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Abstract. We develop a general framework for analyzing KMSstates on C ∗algebras arising from actions of Hecke pairs. We then specialize to the system recently introduced by Connes and Marcolli and classify its KMSstates for inverse temperatures β ̸ = 0,1. In particular, we show that for each β ∈ (1,2] there exists a unique KMSβstate.
VON NEUMANN ALGEBRAS ARISING FROM BOSTCONNES TYPE SYSTEMS
, 2009
"... We show that the KMSβstates of BostConnes type systems for number fields in the region 0 < β ≤ 1, as well as of the ConnesMarcolli GL2system for 1 < β ≤ 2, have type III1. This is equivalent to ergodicity of various actions on adelic spaces. For example, the case β = 2 of the GL2system ..."
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Cited by 6 (5 self)
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We show that the KMSβstates of BostConnes type systems for number fields in the region 0 < β ≤ 1, as well as of the ConnesMarcolli GL2system for 1 < β ≤ 2, have type III1. This is equivalent to ergodicity of various actions on adelic spaces. For example, the case β = 2 of the GL2system corresponds to ergodicity of the action of GL2(Q) on Mat2(A) with its Haar measure.
BOSTCONNES SYSTEMS ASSOCIATED WITH FUNCTION FIELDS
"... Abstract. With a global function field K with constant field Fq, a finite set S of primes in K and an abelian extension L of K, finite or infinite, we associate a C∗dynamical system. The systems, or at least their underlying groupoids, defined earlier by Jacob using the ideal action on Drinfeld mod ..."
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Abstract. With a global function field K with constant field Fq, a finite set S of primes in K and an abelian extension L of K, finite or infinite, we associate a C∗dynamical system. The systems, or at least their underlying groupoids, defined earlier by Jacob using the ideal action on Drinfeld modules and by ConsaniMarcolli using commensurability of Klattices are isomorphic to particular cases of our construction. We prove a phase transition theorem for our systems and show that the unique KMSβstate for every 0 < β ≤ 1 gives rise to an ITPFIfactor of type IIIq−βn, where n is the degree of the algebraic closure of Fq in L. Therefore for n = + ∞ we get a factor of type III0. Its flow of weights is a scaled suspension flow of the translation by the Frobenius element on Gal(F̄q/Fq).
ON A CLASS OF SUBGROUPS OF R ASSOCIATED WITH SUBSETS OF PRIME NUMBERS
"... Abstract. A subgroup G of (R,+) is called representable if there exists a set of prime numbers E such that G = G(E) = ¨t ∈ R;∑p∈E p−1 sin2(t log p) <∞©, or equivalently if it coincides with Connes ’ modular Tgroup associated to a certain ITPFI factor. By our previous work, {0}, R and the cyclic ..."
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Abstract. A subgroup G of (R,+) is called representable if there exists a set of prime numbers E such that G = G(E) = ¨t ∈ R;∑p∈E p−1 sin2(t log p) <∞©, or equivalently if it coincides with Connes ’ modular Tgroup associated to a certain ITPFI factor. By our previous work, {0}, R and the cyclic subgroups of R are representable. Moreover, this class of subgroups is closed under homotheties. It also has the important feature of separating countable subgroups of R from countable subsets of their complements, that is for any countable subgroup H ⊂ R and any countable subset Σ of the complement of H in R there exists a representable group Γ which contains H and does not intersect Σ. In this paper we define and study a natural topology on the space of representable subgroups. This space coincides with the set of equivalence classes of the relation E1 ∼ E2 if and only if G(E1) = G(E2), where E1 and E2 are infinite sets of prime numbers. The structure of the homeomorphisms of this space and the cohomology of a certain natural sheaf are being investigated. A stronger version of the separation property is derived as a corollary of the vanishing of the first cohomology group on certain open sets.
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"... Ergodicity of the action of the positive rationals on the group of finite adeles and the BostConnes phase transition theorem ..."
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Ergodicity of the action of the positive rationals on the group of finite adeles and the BostConnes phase transition theorem
ERGODICITY OF THE ACTION OF K ∗ ON AK
"... ABSTRACT. Connes gave a spectral interpretation of the critical zeros of zeta and Lfunctions for a global field K using a space of square integrable functions on the space AK/K ∗ of adele classes. It is known that for K = Q the spaceAK/K ∗ cannot be understood classically, or in other words, the a ..."
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ABSTRACT. Connes gave a spectral interpretation of the critical zeros of zeta and Lfunctions for a global field K using a space of square integrable functions on the space AK/K ∗ of adele classes. It is known that for K = Q the spaceAK/K ∗ cannot be understood classically, or in other words, the action ofQ ∗ onAQ is ergodic. We prove that the same is true for any global field K, in both the number field and function field cases.