The sumset 
of the sets of integers 
,
, ..., 
, where 
,
is the set of all integers of the form 
, where 
is a member of 
for 
,
2, ..., 
.
The same definition applies to subsets of any additive group, in particular to sets of
real numbers. In this setting, the sum-product
problem compares the cardinalities of the
sumset 
and the product
set 
(Zhang et al. 2026).
See also
Ostmann Conjecture,
Sum-Free
Set,
Sum-Product Problem
Portions of this entry contributed by Tony
Noe
Explore with Wolfram|Alpha
References
Nathanson, M. B. Additive Number Theory: Inverse Problems and the Geometry of Sumsets. New York: Springer-Verlag,
p. 1, 1996.Zhang, K.; Zeng, Y.; and Wang, Y. "Ideal-Smooth
Sets and an Explicit Upper Bound for the Real Sum-Product Exponent." 5 Oct 2026.
https://arxiv.org/abs/2610.05725.Referenced
on Wolfram|Alpha
Sumset
Cite this as:
Weisstein, Eric W., with contributions by Tony Noe. "Sumset." From MathWorld--A Wolfram Resource.
https://mathworld.wolfram.com/Sumset.html
Subject classifications
