On Nov. 30, the suspect tried to enter a home on Potters Court in Great Neck through a rear sliding door at about 2:30 a.m., then ran away, police said.
The assailant did not steal anything, but all the victims who had contact with him were female, police said.
"The offender may be a thrill-seeker with a sexual motive, but that remains unclear at this time," said Det. Lt. Kevin Smith, the police department's chief spokesman. "While there was contact, it did not escalate to a point where a sex crime took place. The subject was fought off or scared off."
Cops are expanding their marked and unmarked patrols to catch the perpetrator or perpetrators, he said.
"We stress that the community lock windows and doors, set alarms and have outdoor lighting. Call at the first sign of suspicious persons," Smith said.
Police also said the same man could have been the intruder Dec. 15 who put a knife to the throat of a woman, 61, who was lying in bed in a home on Georgian Lane in Great Neck. That woman wrestled away the knife and freed herself after the suspect tried to hold her down in her bed, then chased the man out of her home, police said.
Police think he might have broken into two other homes on the same morning, both on Lighthouse Road in Kings Point. Also, police said, the man could be the intruder in a Dec. 12 early morning break-in of a home on West Shore Road in Great Neck.
"It may be related," a police spokeswoman said Thursday. "He may be the suspect in the other incidents. We want the sketch out there to see if people can identify him."
The suspect is about 40, wearing a baseball cap, blue jeans, dark brown shirt and a blue jacket, police said.
Police are once again canvassing the communities where they believe the incidents occurred, Smith said.
He said detectives were also scrutinizing past field interviews, old case reports and the files of offenders known to commit similar crimes.
With Matthew Chayes