Let a1=1 and S_1=1. For n>=2, choose a_n as the least integer greater than a_(n-1) such that |S_(n-1)+exp(-i log a_n)|<|S_(n-1)|, and set S_n=S_(n-1)+exp(-i log a_n). Does this choice exist for every n? If it does, do |S_n| and a_(n+1)/a_n converge, with a positive limit L for the norms and ratio exp(pi-2 arccos(1/(2L)))?