Totally positive elements with $m$ partitions exist in almost all real quadratic fields

TL;DR

The paper proves that for real quadratic fields $K$, there exists an element with exactly $m$ partitions $p_K(\alpha)=m$ in density $1$ of discriminants, highlighting a sharp contrast with the global partition asymptotics. It achieves this by leveraging indecomposable elements, their continued-fraction description, and a detailed analysis of representations $\alpha=e\beta_j+f\beta_{j+1}$; it bounds norms of such elements and classifies cases with $p_K(\alpha|\mathcal{I})=2$, culminating in a complete description for $m\le 7$, including the explicit set $\mathcal{D}(m)$ where no element with $m$ partitions exists. The approach combines Dress–Scharlau indecomposables, convergents/semiconvergents from continued fractions, and combinatorial partition reasoning to derive density results and precise norm bounds. Overall, the work advances understanding of partition ranges in number fields, with concrete criteria for when specific partition counts occur and a complete listing for small $m$, along with sharp asymptotic and density conclusions.

Abstract

In this paper, we study partitions of totally positive integral elements $α$ in a real quadratic field $K$. We prove that for a fixed integer $m \geq 1$, an element with $m$ partition exists in almost all $K$. We also obtain an upper bound for the norm of $α$ that can be represented as a sum of indecomposables in at most $m$ ways, completely characterize the $α$'s represented in exactly $2$ ways, and subsequently apply this result to complete the search for fields containing an element with $m$ partitions for $1 \leq m \leq 7$.

Theorems & Definitions (69)

• Theorem 1.1
• proof
• Theorem 1.2
• proof
• Lemma 2.1: HK
• Theorem 2.2: HK
• Theorem 2.3: HK
• proof
• Lemma 3.1
• proof