Multiple ergodic averages for three polynomials and applications

Abstract

We find the smallest characteristic factor and a limit formula for the multiple ergodic averages associated to any family of three polynomials and polynomial families of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet l 1 p comma l 2 p comma ellipsis comma l Subscript k Baseline p EndSet"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>l</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mi>p</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{l_1p,l_2p,\ldots ,l_kp\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We then derive several multiple recurrence results and combinatorial implications, including an answer to a question of Brown, Graham, and Landman, and a generalization of the Polynomial Szemerédi Theorem of Bergelson and Leibman for families of three polynomials with not necessarily zero constant term. We also simplify and generalize a recent result of Bergelson, Host, and Kra, showing that for all <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>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\varepsilon &gt;0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and every subset of the integers <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda"> <mml:semantics> <mml:mi mathvariant="normal"> Λ </mml:mi> <mml:annotation encoding="application/x-tex">\Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the set <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet n element-of double-struck upper N colon d Superscript asterisk Baseline left-parenthesis normal upper Lamda intersection left-parenthesis normal upper Lamda plus p 1 left-parenthesis n right-parenthesis right-parenthesis intersection left-parenthesis normal upper Lamda plus p 2 left-parenthesis n right-parenthesis right-parenthesis intersection left-parenthesis normal upper Lamda plus p 3 left-parenthesis n right-parenthesis right-parenthesis right-parenthesis greater-than left-parenthesis d Superscript asterisk Baseline left-parenthesis normal upper Lamda right-parenthesis right-parenthesis Superscript 4 Baseline minus epsilon EndSet"> <mml:semantics> <mml:mrow> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.2em" minsize="1.2em">{</mml:mo> </mml:mrow> </mml:mstyle> <mml:mi>n</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> <mml:mo> : </mml:mo> <mml:msup> <mml:mi>d</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.2em" minsize="1.2em">(</mml:mo> </mml:mrow> </mml:mstyle> <mml:mi mathvariant="normal"> Λ </mml:mi> <mml:mo> ∩ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> Λ </mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ∩ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> Λ </mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ∩ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> Λ </mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.2em" minsize="1.2em">)</mml:mo> </mml:mrow> </mml:mstyle> <mml:mo>&gt;</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>d</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> Λ </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mn>4</mml:mn> </mml:msup> <mml:mo> − </mml:mo> <mml:mi> ε