On $n$-ADC integral quadratic lattices over algebraic number fields

In the paper, we extend the ADC property to the representation of quadratic lattices by quadratic lattices, which we define as n -ADC-ness. We explore the relationship between n -ADC-ness, n -regularity, and n -universality for integral quadratic lattices. Also, for n\ge 2 , we give necessary and sufficient conditions for an integral quadratic lattice over arbitrary non-archimedean local fields to be n -ADC. Moreover, we show that over any algebraic number field F , an integral \mathcal{O}_{F} -lattice with rank n+1 is n -ADC if and only if it is \mathcal{O}_{F} -maximal of class number one.