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Caroline Wozniacki: The world at her feet, but there's one frontier to cross

Great Dane is fed up with being asked whether she deserves to be the world No 1 – she just wants to win her first Grand Slam, starting in Paris today. Paul Newman meets Caroline Wozniacki

Imagine the conversation as the neighbours meet outside their front doors. "Hi, good to see you. I noticed you won yet again last week. You're obviously the best player in the world right now."

"Well that's kind of you to say that. You also won, didn't you? So do you enjoy being world No 1?"

The residents of Monte Carlo have become accustomed to living cheek-by-jowl with celebrities, but it is still remarkable that the world's best female tennis player and her male counterpart are next-door neighbours. Caroline Wozniacki, the world No 1, and Novak Djokovic, unbeaten in 2011 and the game's best player on current form, are good friends too.

"We see each other quite a bit," Wozniacki said. "We've been teasing each other. I think I've had a very good season so far, but as I look over my shoulder I see him and he hasn't lost a match yet. I tell him: 'I'm trying to keep up with you, but it's not easy.' He says: 'But you're the No 1 and I'm not yet.' I'm like: 'Yeah, but if you keep up this work you will be soon'."

While the world rankings say that Wozniacki is No 1, the 20-year-old Dane knows she has work to do if she is to be viewed in the same light as her neighbour. Forget Djokovic's remarkable 39-match winning run: this is all about the Serb's two Grand Slam titles.

Wozniacki, who has yet to make her Grand Slam breakthrough, is so used to being asked about whether she deserves to be world No 1 that she began a press conference at this year's Australian Open by answering the question before it was even put to her.

As she sets out on her latest mission at this week's French Open, it is clear that the subject bores rather than irritates her. "My main targets this year are the Grand Slams," she said. "I'd like to win one, but it's not a catastrophe if I don't. I'm just enjoying being on the circuit and playing."

At least going home to Monte Carlo offers an escape from the questions. "I like being able to wake up in the morning, go down to the grocery store, just put my hair up, put some jogging clothes on and nobody really notices," she said. "I think people know who I am but everybody does their own thing there."

It would be wrong to suggest Wozniacki is uncomfortable with her fame. Chosen by Stella McCartney to be the public face of the designer's range of sportswear, she clearly enjoys her lifestyle and the opportunities it brings. She is not the sort to spend all her time on tour either on the tennis court or at a hotel. When in Los Angeles she took the chance to call in on David Beckham – Wozniacki's father played football for Poland and she is a big Liverpool fan – and in the United Arab Emirates earlier this year she met up with Sergio Garcia at the Dubai Desert Classic.

Not that Wozniacki is interested only in celebrities. In Thailand last year she took time out to visit a Buddhist temple. "We went to feed the monks. It's a tradition. You go at 6.40 in the morning, just at sunrise, go on the beach and the monks come. You give them some rice and some water. It should bring you good luck. I had to get up very early but it was a great experience.

"I think it's important to see the cities and the countries where you are. I enjoy it. If it's just the tennis and the hotel you could be anywhere. For me it's very important to experience different cultures, to meet new people. I feel very privileged to be able to play the sport that I love – and to be good at it – and travel to these amazing countries."

She is a good linguist. Polish was her first language, her parents having moved when her father joined a Danish club in the late 1980s, but she was brought up as a Dane. She learned French and English at school, family holidays helped her to understand Russian and she has been teaching herself Spanish.

Her career so far has been a complete contrast to that of her predecessor as world No 1, Serena Williams, who usually turns up the volume at Grand Slam events but makes little noise for the rest of the year. Wozniacki played more matches (79) than any other player on the tour last year, winning six titles.

The Grand Slam will surely come. She lost to Kim Clijsters at the 2009 US Open in her only final but has since reached two semi-finals. In Melbourne earlier this year she had a match point to reach the final but lost to Li Na. She admitted it took a while to recover from that – "I didn't put tennis on the TV afterwards and it was only after four or five days back on the practice court that I felt OK" – though she bounced back in typical fashion by winning three of her next five tournaments, in Dubai, Indian Wells and Charleston.

The critics say Wozniacki's biggest weakness is that for all her great athleticism she lacks a killer shot, to which the Dane has a neat response. "If I don't have a weapon, then what do the others have?" she asked. "Since I'm No 1, I must be doing something right. I think people who say that aren't actually criticising me. I think it's the other players who should be offended."