This post continues my series on favorite theorems of the 21st century. For an overview of the categories and earlier selections, see this post.
My choice for 2001 in Number Theory is Chang’s resolution of a conjecture of Erdős and Szemerédi predicting that a finite set of integers cannot simultaneously have “few” subset sums and “few” subset products. More precisely, they conjectured a superpolynomial lower bound on the combined size of the sets of simple sums and simple products of distinct integers.
Let be a set of distinct integers. Define the sets of simple sums and simple products by Thus consists of all subset sums of , and of all subset products. Set where denotes the cardinality of a set.
Erdős and Szemerédi (Erdős and Szemerédi 1983) conjectured that and cannot both be small. In quantitative terms, they predicted that grows faster than any fixed power of ; that is, is superpolynomial in .
In November 2001, Chang (Chang 2003) confirmed this conjecture with a strikingly precise estimate.
The exponent grows without bound as , so the lower bound is genuinely superpolynomial.
A notable feature of the proof is its direction of attack. A common strategy in sum–product problems is to assume the sum set is small and deduce that the product set must then be large. Chang reverses this logic: assuming the product set is small, she shows that the sum set must necessarily be large. This shift in perspective turns out to be decisive.
Moreover, the result is essentially sharp. It is known that there exists a constant such that so Chang’s theorem determines the correct order of growth up to a constant factor in the exponent.
The analogue of Theorem for general (not necessarily simple) sums and products remains far more difficult and is still open.
For sets , define For a positive integer , define the -fold sumset and product set by
Erdős and Szemerédi (Erdős and Szemerédi 1983) conjectured that for every there exists such that In other words, repeated addition or repeated multiplication must lead to near-maximal growth.
As partial progress toward , Bourgain and Chang (Bourgain and Chang 2004) proved in 2004 that for every integer there exists such that for every nonempty finite set .
Even the case of remains open. In this case the conjecture predicts This is the celebrated Erdős–Szemerédi sum–product conjecture.
In 2009, Solymosi (Solymosi 2009) proved with the exponent in place of . Subsequent work has improved this constant slightly. The current record, due to Bloom (Bloom 2025), establishes the exponent
Chang’s theorem thus stands as a landmark: it completely resolves the “simple” version of the conjecture and achieves the optimal growth rate up to constants in the exponent. At the same time, the general sum–product problem continues to challenge the field, illustrating once again how subtle the interplay between addition and multiplication can be in the integers.