On $p$-adic Transference Theorem

Dual lattice is an important concept of Euclidean lattices. In 2024, Deng gave the definition to the concept of the dual lattice of a $p$-adic lattice from the duality theory of locally compact abelian groups. He also proved some important properties of the dual lattice of $p$-adic lattices, which can be viewed as $p$-adic analogues of the famous Minkowski's first, second theorems and transference theorems for Euclidean lattices. However, he only proved the lower bounds of the transference theorems and Minkowski's second theorem for $p$-adic lattices. The upper bounds are left as an open question. In this paper, we prove the upper bounds of the transference theorems and Minkowski's second theorem for $p$-adic lattices. We then prove that the dual basis of an orthogonal basis is also an orthogonal basis with respect to the maximum norm.