With a shot at claiming the No. 1 ranking, Amelie Mauresmo engaged Elena Dementieva in a contest called, “Anything you can do, I can do worse.”Mauresmo allowed an aching, ailing and double-faulting Dementieva to come back from a set and a break down to win their miscue-marred U.S. Open quarterfinal 4-6, 6-4, 7-6 (1) Tuesday.Dementieva was treated for a strained groin muscle in the first set and for stomach problems in the third, then needed IV fluids for dehydration afterward.”I had some occasions I didn’t take. So I am pretty disappointed with that and a little bit angry with myself,” Mauresmo said. “I just feel very, very disappointed.”How disappointing was the tennis?They combined for 24 double-faults, 82 unforced errors, 36 break points, and 14 service breaks. And all that poor serving wasn’t exactly the result of going for it: Dementieva had zero aces, Mauresmo three.”Her serve is maybe a little weak, but then she’s very quick and very aware that she’s going to have to play a nice shot after,” Mauresmo said. “And that she did pretty well today.”A victory would have guaranteed No. 2 Mauresmo of leading the rankings for the first time. That’s because current No. 1 Justine Henin-Hardenne, the defending Open champion, lost to Nadia Petrova on Monday night.In the wacky world of the computer rankings, though, Mauresmo still will climb to the top despite her loss, unless Lindsay Davenport wins the tournament. That would make Mauresmo only the second woman to be No. 1 without winning a major.She said she wasn’t aware of the chance to secure the top spot Tuesday.In the semifinals, No. 6 Dementieva will face No. 3 Serena Williams or No. 8 Jennifer Capriati, who played Tuesday night before Andy Roddick took on Tommy Robredo.”I’m really looking forward to my next match,” said Dementieva, who also reached the U.S. Open’s final four in 2000. “Both Jennifer and Serena are great players, and I will try to play my best. I’m just so happy to be in the semifinals.”Davenport plays 62nd-ranked Shinobu Asagoe in a quarterfinal Wednesday, when the highlight is top-ranked Roger Federer against two-time Open champ Andre Agassi.In men’s fourth-round action Tuesday, 2001 Open winner Lleyton Hewitt beat Karol Beck 6-4, 6-2, 6-2 for his 14th win in a row, and Tommy Haas beat Tomas Berdych 7-6 (6), 6-1, 7-5, erasing three set points in the tiebreaker.”I’m playing pretty solid tennis, day in, day out,” said Hewitt, who hasn’t dropped a set.Dementieva showed serious fortitude down the stretch. She came out with her left thigh heavily swathed in white tape, then took an injury timeout trailing 5-4 in the first set to get treatment.Mauresmo twice went ahead by a set and a break — the same lead the Frenchwoman wasted against Williams in the Wimbledon semifinals.In the ninth game of the last set, Dementieva began pausing between points. At the next changeover, she took another timeout, this time because of nausea. Later, she tried to cool off in the 80-degree heat by pouring water on her neck.The double-faults just kept coming, 15 in all, including two straight to hand Mauresmo a 4-3 lead in the third set. Dementieva’s long had trouble with that: After hitting 10 in her loss to Anastasia Myskina at Paris in the first all-Russian major final, she said between tears, “I don’t know how to serve.”Each toss is a bit of an adventure, some so far from her body that she needs to lean awkwardly to make contact. Desperate to simply put balls in play, she hit some slower than 60 mph Tuesday — winning one point that began with a 57 mph softball.If her problem is mechanical, Mauresmo’s might be mental. She’s acknowledged having problems with feeling pressure, particularly in front of her countrymen at Roland Garros, and worked with a psychologist.She’s been a quarterfinalist at 10 of the last 11 majors without ever playing in a championship match. Mauresmo’s breakthrough came at the 1999 Australian Open, where she reached the final, but she’s earned a reputation for losing the big ones since.As the match wore on, Mauresmo grew tighter and tighter. By late in the match, when Dementieva’s movement was limited, Mauresmo persisted in hitting returns and other shots right at her weary opponent.Mauresmo lost the last six points, including a forehand return 5 feet long on a 72 mph offering, a forehand that sailed long, and a backhand that flew wide to end it.”I feel I lost the match,” Mauresmo said.