What's wrong with "I am really thank you for ..." and how could I improve it?
I know it's not correct to write "I am really thank you for your gift". Could you explain why it's wrong here and suggest me some ways to improve this sentence. Thank you very much!

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to thank is a verb. You would have to change it to thankful, which is an adjective. You are describing your feelings about the gift.

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I am really thankful for the gift. 
I'm really thankful for the gift.

Thankful, Thank You and Thanks are all different ways to express gratitude

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Dear Minhthi1810
I haven't heard any native people said the sentence which was like you
And "Thank" is a verb, you have to say " I thank you" but not" I am thank you"( or sth like that)
But in the formal English, they usually drop the subject and I think you should correct like this: "Thanks (or thank you) for you gift."
The way to improve: I think you should always find out if the word is a verb, noun or a adjective before writing a sentence.
Claire

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to thank is a verb. I think u could say I really appreciate your gift.