Note on a certain sums of integer parts (Q2777525)

Let \(l,p\) be odd primes and \(H_0\) a subgroup of \(({\mathbb Z}/p^n{\mathbb Z})^*\) of index \(l\). A subset \(T_i\) of a coset \(H_i\) of \(H_0\) in \(({\mathbb Z}/p^n{\mathbb Z})^*\) is called a semisystem (in \(H_i\)) if for each \(x\in H_i\) exactly one of the residue classes \(x\), \(-x\) belongs to \(T_i\). Further, let \(\varepsilon =1\) if \(2\nmid a\), and \(\varepsilon =0\) if \(2|a\). Define \(g(a,i)=\sum _{z\in T_i} (\lfloor a\varepsilon z/p^n\rfloor -\lfloor \varepsilon z/p^n\rfloor)\) and \(G\) as the set of all \(a\in ({\mathbb Z}/p^n{\mathbb Z})^*\) with \(g(a,i)=g(a,j)\pmod {2}\) for all \(i,j\in I\) (here \(\lfloor \cdot \rfloor \) denotes the Gauss brackets). Let \(K\) be a real number field of prime conductor \(p\) and \([K:{\mathbb Q}]=l\). The author proves that if \(2\) is a primitive root modulo \(l\) and \(2\) is not an \(l\)th power modulo \(p\), then \(G=({\mathbb Z}/p^n{\mathbb Z})^*\) if and only if \(h_K\) is even.