Regular sets in residue classes (Q1282531)

Let \(E_m\) denote a complete set of residues mod \(m\), \(A\) a subset of \(E_m\) of cardinality \(| A| \). Call the set \(A\) a basis of \(E_m\) of order \(k=k(A)\) if \(k\) is the least positive integer such that all the equations \(x_1+\cdots+x_k\equiv l\mod m\), \(l\in E_m\), are soluble with some \(x_i\in A\), \(i=1\), \dots, \(k\). Finally, a basis is regular modulo \(m\) if \(k\leq c(\log m)(\log | A|)^{-1}\) for some absolute \(c\geq 1\). The function \(k=k(A)\) is the analog of the function \(G(k)\) in Waring's problem and of course it is of interest to know some explicit regular sets. This paper reviews some known results, including the existence of regular sets mod \(p^N\), \(p\) odd prime and \(N\) large, and also in the more general setting of finite groups. The main new result is the proof that if \(p\) is an odd prime, \(\varepsilon>0\) is fixed and \(N\) is large, then for \(m=\varphi(p^N)\) and \(g\) a primitive root mod \(p^N\), the set \(A=\{\text{ind}_g x:\) \((x,p)=1\), \(x=1\), \dots, \([m^\varepsilon]\}\) is regular mod \(m\) with \(k\leq 2(\log m)(\log | A|)^{-1}+3\).