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 construction using real algebraic integers
need not yield sets of ordinary integers.
Zhang et al. (2026a) 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. (2026b) withdrew their claimed near-quadratic bound over fixed number fields because some mathematical derivations require independent verification. That bound is not regarded as established.
