Those who renounce their citizenship are just the tip of the tax avoidance iceberg. Only the very rich can afford this strategy; it's well beyond the means of the average American. But that doesn't mean Americans aren't finding other ways to avoid the tax man. Literally millions of Americans are simply dropping out of the 'official' economy, and opting into the 'black' economy.
Under the harsh regime of Obamanomics, the U.S. labor participation rate has dropped to it's lowest level in decades. Some point to increased dependence on disability payments, food stamps, and other forms of welfare to explain the trend. While this situation is deplorable, it's only half the story. The majority of those who are dropping out of the labor force and simply moving into the 'under the table' cash economy. They don't count as 'officially' employed, and therefore they don't pay any federal taxes.
It's also interesting to note the cash economy is an outgrowth of our broken immigration system. Illegal immigrants have relied on the cash economy for decades, and in fact pioneered the 'black' economy. But it's rapidly going mainstream; the black economy is "trending." Well done, Mr. President.