He uses two measures of inequality as independent variables in separate specifications: the share of income going to the top 10% and the share going to the top 1%. He uses two estimation techniques in all cases: fixed effects (FE) which identifies effects based on changes within states, and the GMM dynamic panel estimator which allows for endogeneity. He consequently runs 4 regressions on each of nine crime categories. I'll here summarize results from his 36 regressions.
Using fixed effect estimation, an increase in the share of income going to the top 10% reduces violent crime, murder, property crime, burglary, and larceny; all results here significant at the 5% level. In none of these 9 specifications does an increase in inequality increase crime; all the rest are statistically insignificant. If we look instead to the share of income going to the top 1%, increases in inequality reduces violent crime, robbery, property crime, burglary, and larceny; again, all significant at the 5% level, with no FE estimation reporting crime increases.
Using the authors' preferred GMM estimator, and now aggregating across both inequality measures, results are pretty weak. Increased inequality correlates with increased motor vehicle theft, but otherwise has mixed and statistically insignificant results; the increase in motor vehicle theft is only significant at the 10% level in the GMM specification using the fraction of income going to the top 1%.
The paper also includes a control for beer and spirit consumption. In some fixed effect estimations, alcohol consumption increases crime rates; in none of the GMM specifications does it have any effect: we get a mix of insignificant increases and decreases in the various crime rates with increased alcohol consumption.
So it seems not unreasonable to conclude that inequality does not correlate with increased crime rates in the United States, and that it may reduce crime. While the paper's abstract notes that there's perhaps a link between the top 10% income share and motor vehicle theft, that's one significant crime increase out of 36 specifications at the 5% threshold; I wouldn't have put that result in the abstract. Because of, well, this: