As with many other things that money and gentrification have scrubbed from the city, the number of these apartments is shrinking. The roster of rent-controlled apartments, the ones with the most restrictions on rent, has shrunk to 40,000, according to the State Division of Housing and Community Renewal. When these units are emptied, they often become rent stabilized, and after a certain point the apartments can be liberated to the free market. According to the most recent figures available, the number of rent-stabilized units fell to 848,000 in 2008 from 900,000 in 2003. And outside of prime neighborhoods, the restricted rents are not such a great bargain, in many cases barely cheaper than what the apartments could fetch on the open market.

Image The sink is constantly leaking. Credit... Robert Caplin for The New York Times

The deteriorating low-priced apartments still exist partly because landlords have little financial incentive to make renovations or apply for rent increases. Stabilized rents increase by order of the local Rent Guidelines Board, typically around 3 percent a year, while controlled rents can go up only if the landlord files an application for an increase with the state, a laborious process. And the state will not allow a rent increase if an apartment has major housing code violations.

Sherwin Belkin, a Manhattan lawyer who represents owners and developers, said landlords often say it is not worth the time to apply for increases that could translate into just a few dollars more a month.

While it is less true today, rent-regulated apartments have provided some New Yorkers with very comfortable lives. Mr. Burke, who grew up in Ireland on a farm outside Galway, moved to New York in 1964 to be close to his brother upstate and his 100 relatives in Massachusetts. He worked in restaurants and bars and collected decades of tales of celebrity sightings and romances. He moved into his current apartment in 1977, when the rent was $65, and filled his closets with handsome clothes.

Image Mr. Burke says he has spent $13,500 on repairs and $11,000 taking his landlords to court over the conditions. “It’s been hell,” he said. “Now you see, it’s no bargain.” Credit... Robert Caplin for The New York Times

As new owners bought his building and pressed tenants to leave, Mr. Burke held his ground. He was careful to pay his rent by check or money order, not cash, to have a paper trail. But an accident in the Catskills in 1995 left Mr. Burke unable to work and with little spare money. He currently lives on an $1,100 monthly disability check. He finds solace reading historical romance novels about lords and earls by authors like Julia London.