We prove tight probabilistic bounds for the shortest vectors in module lattices over number fields using the results of arXiv:2308.15275. Moreover, establishing asymptotic formulae for counts of fixed rank matrices with algebraic integer entries and bounded Euclidean length, we prove an approximate Rogers integral formula for discrete sets of module lattices obtained from lifts of algebraic codes. This in turn implies that the moment estimates of arXiv:2308.15275 as well as the aforementioned bounds on the shortest vector also carry through for large enough discrete sets of module lattices.

翻译：利用arXiv:2308.15275的研究结果，我们证明了数域上模格中最短向量的紧概率界。此外，通过建立具有代数整数条目且欧几里得长度有界的固定秩矩阵计数的渐近公式，我们证明了从代数码提升得到的离散模格集合的近似罗杰斯积分公式。这进而表明，arXiv:2308.15275中的矩估计以及前述关于最短向量的界，对于足够大的离散模格集合同样成立。