Infinite sumsets in sets with positive density
Bryna Kra, Joel Moreira, Florian Richter, Donald Robertson
Abstract
Motivated by questions asked by Erdős, we prove that any set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A subset-of double-struck upper N"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo> ⊂ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">A\subset \mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with positive upper density contains, for any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k element-of double-struck upper N"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">k\in \mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , a sumset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B 1 plus midline-horizontal-ellipsis plus upper B Subscript k"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mo> ⋯ </mml:mo> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">B_1+\cdots +B_k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B 1"> <mml:semantics> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">B_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , …, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Subscript k Baseline subset-of double-struck upper N"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo> ⊂ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">B_k\subset \mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are infinite. Our proof uses ergodic theory and relies on structural results for measure preserving systems. Our techniques are new, even for the previously known case of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k equals 2"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k=2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .