For which real quadratic fields is Kim's octonary form universal?

Abstract

Let $K=\mathbb{Q}(\sqrt{D})$ with $D>1$ squarefree, and let $\varepsilon_+$ be the totally positive fundamental unit of $\mathcal{O}_K$. B. M. Kim proved in 2000 that the octonary diagonal form \[ f=x_1^2+\cdots+x_4^2+\varepsilon_+(x_5^2+\cdots+x_8^2) \] is universal over $\mathcal{O}_K$ whenever $D=n^2-1$ is squarefree. We complete Kim's result to an if-and-only-if classification: $f$ is universal if and only if $D=n^2-1$ for some $n\ge2$, or $D=n^2-4$ for some odd $n\ge3$, in both cases subject to squarefreeness. The second family appears to be new in this context and contains $\mathbb{Q}(\sqrt{5})$ at $n=3$ as a degenerate boundary case, recovering Maaß's classical exceptional three-square phenomenon. Equivalently, $f$ is universal over $\mathcal{O}_K$ if and only if the Blomer--Kala invariant $M_D$ equals $1$; for the two stated families we have $M_D=1$, so the Blomer--Kala universal $8M_D$-variable construction specializes exactly to $f$. The converse leverages a continued-fraction involution $τ(γ)=\varepsilon_+γ'$ together with a closed formula in convergent coordinates, a three-consecutive-square-values lemma for primitive quadratic polynomials of positive fundamental discriminant, and an even-root exclusion lemma derived from complete-quotient norm bounds.