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1-Let me know if you are busy on the night.

What does 'on the night' mean in this sentence?

Can one say:

2-Any day this week you find out that you are going to be free on the night, give me a call.

3-Any day this week you find out that you are going to be free on that night, give me a call.

4-Any day this week you find out that you are going to be free on the night of that day, give me a call.
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Sorry, but it doesn't mean 'Let me know if you are busy at night.'. Your sentence here means nights in general, and the original refers to one specific night.

'busy on the night' - busy on the evening already being discussed.
Comments  
It means the same as :
Let me know if you are busy at night.
I find this more appropriate.

Sentences 2,3 and 4 are the same but 2 and 3 are more specific than 1.
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 nona the brit's reply was promoted to an answer.
2-Any day this week you find out that you are going to be free on the night, give me a call.

3-Any day this week you find out that you are going to be free on that night, give me a call.

4-Any day this week you find out that you are going to be free on the night of that day, give me a call.

these don't work with 'on the night' because you aren't discussing a specific night. Your sentences would be better expressed as

Any day this week you find out you are going to be free THAT night, give me a call. (so if it is Monday, and the person knows they will be free Monday night, call).
Thank you. You made it clear. I am glad I am now able to see why.
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