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I'm not sold that it means "difficult to exist/rare", instead of "there is a difficulty." It makes sense to think of thanking somebody as also apologizing for having them deal with a difficulty.



It’s a straightforward form of 複合形容詞 (compound adjective), formed from verb stem + adjective. For instance:

- 読み難い: yomu (to read) + nikui (difficult) -> yominikui: “difficult to read”

- 分かり易い: wakaru (to understand) + yasui (easy) -> wakariyasui: “easy to understand”

Applying this construction:

- 有り難い: aru (to be) + katai (difficult) -> arigatai: literally “difficult to be”




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