The density finite sums theorem

Abstract For any set $A$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> of natural numbers with positive upper Banach density and any $k\geqslant 1$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>1</mml:mn> </mml:math> , we show the existence of an infinite set $B\subset \mathbb{N}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>B</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>N</mml:mi> </mml:math> and a shift $t\geqslant 0$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>t</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>0</mml:mn> </mml:math> such that $A-t$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> <mml:mo>−</mml:mo> <mml:mi>t</mml:mi> </mml:math> contains all sums of $m$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>m</mml:mi> </mml:math> distinct elements from $B$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>B</mml:mi> </mml:math> for all $m\in \{1,\ldots ,k\}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>m</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>{</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> <mml:mo>}</mml:mo> </mml:math> . This can be viewed as a density analog of Hindman’s finite sums theorem. Our proof reveals the natural relationships among infinite sumsets, the dynamics underpinning arithmetic progressions, and homogeneous spaces of nilpotent Lie groups.