1. An archeologist must be an expert in (…on, at?) finding old tiles. [What is the correct preposition in combination with "expert" and the gerund?]

2. Because the technology will be brilliant in 4004, the spaceships will fly faster than the light. [Is it O.K. to use "Because" at the beginning of the sentence?]

3. Lola has no time. Has she? [oder Hasn´t she?]

Thanks for help!

1. Expert at. You can be "expert in" a field or area of specialization, but when it comes to individual actions, the gestalt of containment "in" something is gone. Being an "expert on" something usually connects with topic names, "expert on plant health" and so on. As an expert in the field of botany, she was a leading expert on plant health, and was expert at spotting aphids hiding under leaves.

2. People sometimes say that sentences ought not to be started with "because," but it's commonly done except in the most formal or artistically classical formats. It is common to use "since" or "as" as a substitute, but neither of those would sound at all strong in your enthusiastic sentence. Also, there isn't really a cause-and-effect relationship between the technology being wonderful in 4004 and the idea that spaceships will then go to warp 7. The faster-than-light flight will be a manifestation or example of ingenious technology, not an effect of it. So where do you go? Well, you have an extra "the" in "the light" that doesn't belong in an English sentence, and also, technology, as an abstract collection of working processes, can't be "brilliant" except in British slang, so here's what you might get mutatis mutandis: Technology in 4004 will be fantastically advanced -- spaceships will fly faster than light!

Someone else may be able to come with a stronger version; that one's still a bit tepid. The word "dazzling" can sometimes be used in rhetoric of high enthusiasm, but I am hard pressed to combine it with "technology." In Dutch we could indeed exactly say "brilliant" (schitterend) in this context, and perhaps your German has an equivalent term, but there is no exact English translation of that general term of high praise. Brilliance is ascribed to people, to bright objects, to plans and to performances or productions, but not to abstract collectives. You can't say that Asians have brilliant societies or that the social sciences have brilliant fundamental methods.

3. Lola has no time, has she? If you used "hasn't she," that would make a banned double-negative.
Johnno, you are perfectly correct in what you have written about question tags.
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Dear Brattania,

Thanks so much for your detailed explanantion, which gives much more information than in any available dictionary!

Juergen (Germany)
Just a quick note. My understanding is that question tags are the opposite of the statement.

You don´t eat children, do you?

Negative statement -positive tag.

He has some apples, hasn´t he?

Positive statement-negative tag.

Lola hasn´t any time, has she?
Lola has no time, has she?
OR you could go with 'do'
Lloa doesn´t have any time, does she?

Cheers, Johnno
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 taiwandave's reply was promoted to an answer.