During my stay in the United States, I accumulated books and papers. Books and papers can be pretty heavy so it may be very expensive to bring them by plane. For example for half cubic meter of books, I wonder if there is another way. If time is not a matter what is the cheapest way to send books from USA to France?
The cheapest way to ship anything is to get somebody to carry it for you as a favour.
If the amount of stuff is small and/or the friend is a close/good one the price will be zilch.
If you're a nice person and especially if the mule was a friend of a friend or somebody you found on a social site then you could offer them a six pack of their favourite beer, a place to crash while they're in town, or even an amount of money you can afford but which is lower than the cost of using a shipping company would've been.
This is the cheapest way given no other restrictions.
If you have requirements such as needing the stuff within a short period of time, not having enough friends, or having so much stuff that part would need to be carried by each of several people, then that would be a whole other question.
Note that historically Travel.SE regards shipping that's not a requirement of your trip to be off-topic. Getting specialized gear to a place you'll need it for your adventure would be on-topic. (Climbing gear, hang-glider, off-road vehicle, kayak, etc.)
Note also that historically we tend to allow a lot of questions anyway if they're very specific so that they wouldn't get a whole bunch of correct answers or a whole bunch of guesses and speculative answers. In such cases you'll want to include all the information that's special about your predicament.