NEW YORK — Billionaire Michael Bloomberg won a third term as New York mayor Tuesday in a closer-than-expected race against a Democratic challenger who stoked voter resentment over the way Bloomberg changed the city’s term-limits law so he could stay in office.

With all precincts reporting, Bloomberg, the richest man in New York and founder of the financial information company Bloomberg LP, defeated William Thompson Jr. 51 percent to 46 percent.

The mayor called it a “hard-fought victory in a very difficult year,” and promised that New Yorkers “ain’t seen nothing yet” from him.

“I’m committed to working twice as hard in the next four years as I did in the past eight,” Bloomberg said.

In the days leading up to the election, polls showed Bloomberg with as much as an 18-point lead, an edge so big that critics accused the mayor of overkill in his strategy of bombarding the city with campaign ads.

His actual margin of victory was far smaller than the nearly 20-point blowout he pulled off in 2005.

When all the bills are paid, Bloomberg will probably have spent more than $100 million on his campaign, the most expensive self-financed campaign in U.S. history. Thompson, the city’s comptroller, relied on donations and matching funds for his mayoral bid, and was on track to spend about a tenth of Bloomberg’s staggering total.

“This campaign was about defying conventional wisdom … this campaign was about standing strong, standing tall and never backing down in the face of a formidable challenge,” Thompson said after conceding defeat.

Thompson ran up huge margins in black and Hispanic neighborhoods, winning by a 3-to-1 margin in some districts.