In their 2005 book Sinatra:The Life authors Anthony Summers and Robbyn Swan write extensively (on pages 325-327) about the singer's drinking. They quote Dr. Robert Morse, chairman of the Medical/Scientific Committee of the National Council on Alcoholism and Drug Dependence: "There is little doubt that Sinatra had a serious 'drinking problem.' ... To have such a high tolerance—I see references to consumption of a bottle a day of Jack Daniel's—is the most specific symptom. One can have this kind of tolerance only by becoming an alcoholic. Much of what we know about Sinatra's drinking is consistent with alcoholism."
Anthony Summers contacted me prior to publication. Based on portions of the unpublished manuscript he shared with me, including references to his reported bottle-a-day habit I told him I thought "Sinatra was an alcoholic." He quotes me on page 327 of the published book.
Jack Daniel's currently has a number of commercials on American television featuring Sinatra. One can be viewed here, another here and yet another here.