Stable functions and Følner’s Theorem

We show that if <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> is an amenable group and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A subset-of-or-equal-to upper G"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo> ⊆ </mml:mo> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">A\subseteq G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has positive upper Banach density, then there is an identity neighborhood <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the Bohr topology 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> that is almost contained in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A upper A Superscript hyphen 1"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>-</mml:mtext> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">AA^{\text {-}1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the sense that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B minus upper A upper A Superscript hyphen 1"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mi class="MJX-variant" mathvariant="normal"> ∖ </mml:mi> <mml:mi>A</mml:mi> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>-</mml:mtext> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">B\backslash AA^{\text {-}1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has upper Banach density <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This generalizes the abelian case (due to Følner) and the countable case (due to Beiglböck, Bergelson, and Fish). The proof is indirectly based on local stable group theory in continuous logic. The main ingredients are Grothendieck’s double-limit characterization of relatively weakly compact sets in spaces of continuous functions, along with results of Ellis and Nerurkar on the topological dynamics of weakly almost periodic flows.