Are you on a lease? Then read it. It probably says something about the owner maintaining the property to enable the tenant peaceful enjoyment of the rental property, or words to that effect.
Clearly, odors of rot prevent that. So you need to document your actions in a way that will stand up in court, should it come to that. (Let's hope it won't.)
Send an email with a "read" receipt if your email can do that, or a registered letter, return receipt requested, confirming what you were told. Word it something like this. "I am following up on my conversation with your office [name the person if you know their name] on May 21, 2019 concerning the odors of trash and rotting food entering my apartment at [complete address] from the unit next door at [complete address]. I was assured the problem would be taken care of on May 22, 2019. If this is no longer correct, please advise me of the date of the completion of the work and let me know in what way I will be compensated for the failure to make my unit safe to inhabit."
If they don't act, then you need to get the town's or city's housing inspector involved. They are likely to declare your unit unfit for habitation, which will spur the landlord to act, especially if it involves a fine. But they may also require you to vacate, and that's a pain.