% Shared manuscript packages, notation and theorem environments. \usepackage[T1]{fontenc} \usepackage{lmodern} \usepackage[margin=30mm]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{microtype} \usepackage[numbers,sort&compress]{natbib} \usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref} \hypersetup{ pdftitle={Distinct partial products in prime dihedral groups}, pdfauthor={}, pdfsubject={Combinatorics and group sequencing}, pdfkeywords={dihedral groups, distinct partial products, Graham's rearrangement conjecture, anticoncentration} } \newcommand{\Fp}{\mathbb F_p} \newcommand{\Z}{\mathbb Z} \newcommand{\D}{\mathcal D} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\Span}{span} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} \setlength{\emergencystretch}{2em}