The sum-product problem asks how small both the sumset 
and product
set 
of a large finite set 
can be. The ambient field matters,
and a bound for real sets need not hold for sets
of integers.
Zhang et al. (2026) constructed arbitrarily large finite sets of real algebraic integers satisfying
 |
This gives an explicit upper bound for the real sum-product exponent along an unbounded sequence of cardinalities. It does not refute the corresponding near-quadratic conjecture for integers. Zhang et al. (2026) report AI assistance with selected computations, while attributing the mathematical ideas and proofs to the authors.
