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Unless the landlord is denying this application based on the applicants being a part of a protected class (age, gender race, disability, etc.) then there should be no issue for the background checks or a withdraw of the lease offer. This is a privately owned apartment and the landlord can rent to whomever he/she pleases as long as any denial is not being made based on the applicant being in a protected class. The landlord can do whatever background check he/she finds appropriate and this is not unlawful. With the facts as you present them, there doesn't sound to be any basis for any applicant to file suit as the landlord is well within his/her rights to perform all the actions described.
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I would not use that as a basis for denial. I would simply state that they did not pass your background check. You do not need to specify anything more than that. If you want to be more specific you could also say that they were not a good fit as a tenant.
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