2021/02/28 by Or Shalom · 14 citations
Mathematics · Psychology · #Abelian group #Ergodic theory #Geology #Graph theory and applications #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #Mathematics #Psychology #Pure mathematics #Statistics #Type (biology) #math.CO #math.DS
paper · pdf · doi:10.1090/tran/8558
published in Transactions of the American Mathematical Society 375(4), 2729-2761 (American Mathematical Society) · 37 pages, 1 figure. final accepted version, to appear in Trans. Amer. Math. Soc. Added a structure result for the Conze-Lesigne factor as a double coset, simplified the proofs in section 3 and added various examples
openalex created_date 2021/03/01 · arxiv created 2021/09/12 · openalex publication_date 2021/09/29 · arxiv updated 2022/01/12 · openalex updated_date 2026/08/05
Let <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> be a countable abelian group. We study ergodic averages associated with configurations of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet a g comma b g comma left-parenthesis a plus b right-parenthesis g EndSet"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"></mml:mo> <mml:mi>a</mml:mi> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>+</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>g</mml:mi> <mml:mo fence="false" stretchy="false"></mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\ag,bg,(a+b)g\</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a comma b element-of double-struck upper Z"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">a,b∈ \mathbb Z</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Under some assumptions on <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> , we prove that the universal characteristic factor for these averages is a factor (Definition 1.15) of a <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> -step nilpotent homogeneous space (Theorem 1.18). As an application we derive a Khintchine type recurrence result (Theorem 1.3). In particular, we prove that for every countable abelian group <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> , if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a comma b element-of double-struck upper Z"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">a,b∈ \mathbb Z</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a upper G comma b upper G comma left-parenthesis b minus a right-parenthesis upper G"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>b</mml:mi> <mml:mo> − </mml:mo> <mml:mi>a</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">aG,bG,(b-a)G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis a plus b right-parenthesis upper G"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>+</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">(a+b)G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are of finite index in <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> , then for every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E subset-of upper G"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo> ⊂ </mml:mo> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">E⊂ G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi> ε </mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">ε >0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the set <disp-formula content-type="math/mathml"> <mm