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In this book there are three worlds that are somehow connected together. One world is the Earth as we know it today, one is the Earth in medieval times and one is the Earth in a strange old Egyptian style with futuristic weapons.
The three main characters are the same person each in one of the three worlds, and they have a task that I don't remember. I think they had to accomplish the task together. Sometimes they had the possibility to communicate to each other.
Maybe you're thinking of the Pendragon series? It's a ten-book series, and tracks Bobby Pendragon, a Traveler, as he explores different worlds. He comes from our Earth, from the Second Earth era. He travels to First and Third Earth (among other worlds, called "territories") while fighting Saint Dane (a shapeshifting demon). He had a slightly different challenge on each Earth, trying to prevent Saint Dane from destroying worlds and cultures with different methods, including (Wikipedia) "cultural homogenization, social inequality, discrimination, and totalitarianism." (Spoiler: they don't always win.)
The three main characters would be Bobby, his girlfriend Courtney, and his friend Mark. They spend some time together on different worlds, but for a lot of the series they're doing different things on different worlds, and keep in touch through journals and other methods (my memory of this is a little fuzzy).
Here are some of the covers (from Wikipedia):
(By Source, Fair use, https://en.wikipedia.org/w/index.php?curid=14051197)
