Lattice packing of spheres in high dimensions using a stochastically evolving ellipsoid

Abstract We prove that in any dimension $n$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> there exists an origin-symmetric ellipsoid ${\mathcal{E}} \subset {\mathbb{R}}^{n}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> of volume $c n^{2} $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>c</mml:mi> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> that contains no points of ${\mathbb{Z}}^{n}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Z</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> other than the origin, where $c &gt; 0$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>c</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:math> is a universal constant. Equivalently, there exists a lattice sphere packing in ${\mathbb{R}}^{n}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> whose density is at least $cn^{2} \cdot 2^{-n}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>c</mml:mi> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>⋅</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:math> . Previously known constructions of sphere packings in ${\mathbb{R}}^{n}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> yielded densities of at most $C n \log n \cdot 2^{-n}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> <mml:mi>n</mml:mi> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>⋅</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:math> . Our proof utilizes a stochastically evolving ellipsoid that accumulates at least $c n^{2}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>c</mml:mi> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> lattice points on its boundary, while containing no lattice points in its interior except for the origin.