The scarcity of archaeological evidence that can be reliably dated to the period AD 614–911, on perceived inadequacies of radiometric and dendrochronological methods of dating this period, and on the over-reliance of medieval historians on written sources.
The presence of Romanesque architecture in tenth-century Western Europe. This is taken as evidence that less than half a millennium could have passed since the fall of the Roman Empire, and concludes that the entire Carolingian period, including the existence of the individual known as Charlemagne, is a forgery by medieval chroniclers; or more precisely, a conspiracy instigated by Otto III and Gerbert d'Aurillac.
The relation between the Julian calendar, Gregorian calendar and the underlying astronomical solar or tropical year. The Julian calendar, introduced by Julius Caesar, was long known to introduce a discrepancy from the tropical year of around one day for each century that the calendar was in use. By the time the Gregorian calendar was introduced in AD 1582, Illig alleges that the old Julian calendar "should" have produced a discrepancy of thirteen days between it and the real (or tropical) calendar. Instead, the astronomers and mathematicians working for Pope Gregory had found that the civil calendar needed to be adjusted by only ten days. From this, Illig concludes that the AD era had counted roughly three centuries which never existed.
Regarding the Gregorian reform: It was never intended or purported to bring the calendar in line with the Julian calendar as it had existed in 45 BC, the time of its institution, but as it had existed in 325, the time of the Council of Nicaea, which had established a method for determining the date of Easter Sunday by fixing the Vernal Equinox on March 20 in the Julian calendar. By 1582, the astronomical equinox was occurring on March 10 in the Julian calendar, but Easter was still being calculated from a nominal equinox on March 20. In 45 BC the astronomical vernal equinox took place around March 23rd. Illig's "three missing centuries" thus correspond to the 369 years between the institution of the Julian calendar in 45 BC, and the fixing of the Easter Date at the Council of Nicaea in AD 325.