INTEGRALITY OF AVERAGES OF ROOTS OF UNITY AND PERFECT ISOMETRIES

Abstract We establish a criterion for the integrality of averages of roots of unity and apply it to settle a conjecture regarding the linearity of functions on <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> </mml:math> $\mathbb {Z}_n$ double struck upper Z Subscript n . Specifically, we prove that for any modulus <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:math> $n \ge 1$ n greater than or equals 1 , if <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false" accent="false">→</mml:mo> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> </mml:math> $f: \mathbb {Z}_n \to \mathbb {Z}_n$ f colon double struck upper Z Subscript n Baseline right arrow double struck upper Z Subscript n Baseline is a function whose averages <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:munderover> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:munderover> <mml:msup> <mml:mi>ω</mml:mi> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>b</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:msup> </mml:math> $({1}{/n}) \sum _{x=0}^{n-1} \omega ^{f(x)+bx}$ left parenthesis 1 slash n right parenthesis sigma summation Underscript x equals 0 Overscript n minus 1 Endscripts omega Superscript f left parenthesis x right parenthesis plus b x (where <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>ω</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> <mml:mi>i</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:math> $\omega =e^{2\pi i/n}$ omega equals e Superscript 2 pi i divided by n ) are algebraic integers for all <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>b</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> </mml:math> $b \in \mathbb {Z}_n$</