I am looking for a novel which I read in the mid/late eighties. I was a teenager then, so it might be a YA novel. I read it in German; whether it was translated or a German original I cannot say.
The protagonist, a young man or boy, is at a lake or river with some friends. He is swimming, his friends are on the shore, a radio is playing. He dives, and finds the entrance to a cave. He swims in, surfaces, and since the cave is filled with air, spends some time there examining it. Suddenly it occurs to him that his friends must be worried and possibly thinking that he has drowned, so he returns. But when he resurfaces, no one has noticed that he was gone for so long. The same song is still playing on the radio, so he realizes that no time has passed while he was in the cave. He returns to the cave several times on later occasions, and eventually finds out that it is a portal and can be used to travel to either another world or the future (I cannot remember which).
The second scene (or rather setting) that I remember is the world the portal leads to: The complete surface of this world is covered by metal tiles, and the population lives underground. However there is one island where the ruling class lives, above ground and in luxury. I think the cover of the novel shows the tiled surface of this world, stretching endlessly. One tile might be opened, like a hatch.
Finally I remember that either an advanced alien race or humans from the future have left humanity a message long ago, but they cannot understand it yet because it was encoded using higher mathematics. This was done to ensure that humankind is sufficiently advanced before they can read the message.
It is possible that the three scenes are from two or three separate books, but if so, they are probably part of the same series. This was one of the first science-fiction novels I read, and I would very much like to read it again, so any help is appreciated.