Estimation of Stellar Distances and Refutation of Ptolemy's Model
Gersonides is the only astronomer before modern times to have estimated correctly stellar distances. Whereas all other astronomers put the stars on a rotating sphere just beyond the outer planets, Gersonides estimated the distance to the stars to be ten billion times greater, of the order of 100 light-years (in modern units).
Using data he collected from his own observations Gersonides' refuted Ptolemy's model in what the notable physicist Yuval Ne'eman has considered as "one of the most important insights in the history of science, generally missed in telling the story of the transition from epicyclic corrections to the geocentric model to Copernicus' heliocentric model". Ne'eman argued that after Gersonides reviewed Ptolemy's model with its epicycles he realized that it could be checked, by measuring the changes in the apparent brightnesses of Mars and looking for cyclical changes along the conjectured epicycles. These thus ceased being dogma, they were a theory that had to be experimentally verified, "a la Popper". R. Levi developed tools for these measurements, essentially pinholes and the camera obscura.
The results of his observations did not fit Ptolemy's model at all. Gersonides concluded that the model was no good. He tried (unsuccessfully) to improve on it. That challenge was finally answered, of course, by Copernicus three centuries later, but Gersonides was the only one to falsify the Alexandrian dogma - the first known instance of modern falsification philosophy. Levi also showed that Ptolemy's model for the Lunar orbit, though reproducing correctly the evolution of the Moon's position, fails completely in predicting the apparent sizes of the Moon in its motion. Unfortunately, there is no evidence that the findings had an impact on later generations of astronomers, even though Gersonides' writings were translated and available.
You state that Gersonides calculated that the stars were 10 billion times farther away from Earth than what Ptolemy claimed, i.e. in the order of 100 light years. This should be about 100,000 light years (107,055 to be more precise).
As the mean radius is about 6,370 km this works out to 127,323,954 km. Gersonides however took it to be more than 159 x 1012. So Gersonides works out to 1.012830 exp19 km. As a light year is 9.4608exp13 km it works out to 107,550 light years, which is indeed very close to 10 billion times as far as Ptolemy's distance.
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