On average coherence of cyclotomic lattices (Q6585106)

The paper introduces a new invariant of a lattice, its average coherence: For a lattice \(L\) in Euclidean space, let \(S\) represent the pairs \(\pm v\) of minimal vectors \(v\in L\). Then the average coherence of \(L\) is \N\[\NA(L) := \frac{1}{|S|-1} \max \{ \sum _{v\in S \setminus \{ v_0 \} } \frac{|<v,v_0>|}{\|v\| \|v_0\|} \mid v_0 \in S \} .\N\]\NThe main result of the paper computes the average coherence for cyclotomic lattices. For a natural number \(n\geq 3\) put \(\tau (n)\) to denote the number of divisors of \(n\). Then the average coherence of the \(n\)-th cyclotomic lattice is given by \(\frac{2^{\tau (n)-1}}{n-1} \) if \(n\) is odd and \(\frac{2^{\tau (n)-2}}{n-2} \) if \(n\) is even. The authors relate this notion to the orthogonality defect and product measure and give examples for these values for certain cyclotomic lattices and all irreducible root lattices.