On a problem of Kodama concerning the Hasse-Witt matrix and the distribution of residues (Q1104966)

The problem in the title asks whether for an odd prime f there exist an integer c coprime to f and an integer j such that the following property holds: the least residue of jc n mod f is in the interval \([1, (f-1)/2]\) for all n with \(0\leq n\leq r-1,\) where r is the multiplicative order of c mod f. This problem arose in connection with studies of the rank of the Hasse-Witt matrix for hyperelliptic function fields over finite fields. It is shown that whenever integers c and j with the desired property exist, then necessarily \(r=O(f^{1/2} \log f)\) with an absolute implied constant. The key ingredient of the proof is a bound for exponential sums established by the author [Bull. Am. Math. Soc. 84, 957-1041 (1978; Zbl 0404.65003)].