Representation Functions on the Additive Group of Residue Classes

Abstract

For any positive integer m, let ℤ_m be the additive group of residue classes modulo m. For A ⊆ ℤ_m and

\overline{n}\in {\mathbb{Z}}_m

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\overline{n}\in\mathbb{Z}_{m}$$\end{document}

, let the representation function

R_A(\overline{n})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R_{A}(\overline{n})$$\end{document}

denote the number of solutions of the equation

\overline{n}=\overline{a}+\overline{a^{\mathrm{\prime }}}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\overline{n}=\overline{a}+\overline{a^{\prime}}$$\end{document}

with unordered pairs

(\overline{a},\overline{a^{\mathrm{\prime }}})\in A\times A

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(\overline{a},\overline{a^{\prime}})\in A\times A$$\end{document}

. Let m = 2^αM > 2, where α is a positive integer and M is a positive odd integer. In this paper, the author proves that if M ≥ 3, then there exist two distinct sets A, B ⊆ ℤ_m with ∣A ∪ B∣ = m − 2, A ∩ B = ∅ and

B\ne \frac{\overline{m}}{2}+A

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B\ne{\overline{m}\over{2}}+A$$\end{document}

such that

R_A(\overline{n})=R_B(\overline{n})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R_{A}(\overline{n})=R_{B}(\overline{n})$$\end{document}

for all

\overline{n}\in {\mathbb{Z}}_m

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\overline{n}\in\mathbb{Z}_{m}$$\end{document}

. The author also proves that if M = 1 and A, B ⊆ ℤ_m with ∣A ∪ B∣ = m − 2 and A ∩ B = ∅, then

R_A(\overline{n})=R_B(\overline{n})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R_{A}(\overline{n})=R_{B}(\overline{n})$$\end{document}

for all

\overline{n}\in {\mathbb{Z}}_m

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\overline{n}\in\mathbb{Z}_{m}$$\end{document}

if and only if

B=\frac{\overline{m}}{2}+A

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B={\overline{m}\over{2}}+A$$\end{document}

.

Keywords

Representation function / Residue class

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