The structure of arbitrary Conze–Lesigne systems

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ<!-- Γ --></mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a countable abelian group. An (abstract) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ<!-- Γ --></mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-system <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> - that is, an (abstract) probability space equipped with an (abstract) probability-preserving action of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ<!-- Γ --></mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> - is said to be a <italic>Conze–Lesigne system</italic> if it is equal to its second Host–Kra–Ziegler factor <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Z squared left-parenthesis normal upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">Z</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">X</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {Z}^2(\mathrm {X})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The main result of this paper is a structural description of such Conze–Lesigne systems for arbitrary countable abelian <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ<!-- Γ --></mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, namely that they are the inverse limit of translational systems <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscript n Baseline slash normal upper Lamda Subscript n"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi mathvariant="normal">Λ<!-- Λ --></mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">G_n/\Lambda _n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> arising from locally compact nilpotent groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscript n"> <mml:semantics> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">G_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of nilpotency class <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, quotiented by a lattice <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda Subscript n"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Λ<!-- Λ --></mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\Lambda _n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Results of this type were previously known when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ<!-- Γ --></mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> was finitely generated, or the product of cyclic groups of prime order. In a companion paper, two of us will apply this structure theorem to obtain an inverse theorem for the Gowers <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U cubed left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>U</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">U^3(G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm for arbitrary finite abelian groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

Locations

• Communications of the American Mathematical Society - View
• arXiv (Cornell University) - View - PDF
• Communications of the American Mathematical Society - View
• arXiv (Cornell University) - View - PDF
• Communications of the American Mathematical Society - View
• arXiv (Cornell University) - View - PDF