Xinzhiyuan Report
Editor: Rhino
[Xinzhiyuan Guide] An 18-year-old mathematical problem, previously unsolved, has been broken three times in just 30 days through a collaboration between AlphaEvolve and humans! The sum-difference set exponent θ has been raised from 1.173050 to 1.173077, pushing the ceiling of additive combinatorics to a new height. This "human-machine dance" not only shocked the mathematics community but also heralded a new era of scientific discovery.
The world of mathematics witnesses another miracle!
An 18-year-old unsolved problem has been broken three times in a single month by a collaboration between AI and humans!
Each time, our understanding of what's possible is pushed to new heights.
On June 2, Fan Zheng's latest paper on arXiv once again pushed the record for the sum-difference set exponent θ up by 0.000027, from 1.173050 to 1.173077.
0.000027 – a span barely discernible under a microscope, yet it lifted the ceiling of additive combinatorics by another inch.
Paper link: https://arxiv.org/abs/2506.01896
Such rapid and continuous progress is inseparable from the mutual cooperation between mathematicians and AI (AlphaEvolve).
It can be said that AI is helping humanity advance the scientific frontier!
This breakthrough even amazed Terry Tao, who exclaimed: "To me, this is a fascinating illustration."
Tao believes this demonstrates how highly computer-assisted, moderately computer-assisted, and traditional "paper-and-pencil" methods will interact in future mathematical research.
Each of these paradigms has its advantages and disadvantages.
For example, the current AlphaEvolve finds it extremely difficult to use the asymptotic constructions used in subsequent papers; but on the other hand, without AlphaEvolve's brute-force search, human methods would also find it difficult to discover the entry points for these improvements.
His words inevitably bring to mind Newton's classic quote: "If I have seen further than others, it is by standing upon the shoulders of giants."
However, here the giants now include not only the accumulated wisdom of predecessors but also powerful AI tools like AlphaEvolve.
What are sum-difference sets?
Let's zoom in and delve into the sum-difference set problem that has fascinated mathematicians for decades.
Simply put, it's a game about sets of integers, with the core being the comparison of the sizes of the sum set (A+B) and the difference set (A-B).
Imagine two bags A and B filled with integers:
Sum set (A + B): Pick one from each, add them, and collect all possible results into a set.
Difference set (A − B): Perform the same operation but with subtraction.
For example, if A={1, 2} and B={3, 4}, then A+B={1+3, 1+4, 2+3, 2+4} = {4, 5, 5, 6} = {4, 5, 6} (elements not repeated).
A-B={1-3, 1-4, 2-3, 2-4} = {-2, -3, -1, -2} = {-1, -2, -3}.
The game mathematicians play is to:
Make the sum set small and the difference set large.
The index measuring "largeness" is θ. The higher the limit, the more powerful it is; theoretically, the upper bound for θ is 4⁄3 (≈1.3333).
3 Record Breaks in a Month
How were these three breakthroughs achieved? Let's break them down one by one.
In 2007, Hungarian mathematicians Gyarmati, Hennecart, and Ruzsa constructed a set containing approximately 30,000 elements, fixing the lower bound of θ at 1.14465.
This record stood like an insurmountable peak for 18 years.
However, this record was broken on May 14, 2025.
DeepMind's AlphaEvolve (with Terry Tao's involvement) acted like an explorer with boundless energy and a unique perspective, performing a "breadth scan" through a vast space of possibilities using a novel evolutionary algorithm.
The result was astounding: AlphaEvolve found a new set containing 54,265 elements, immediately raising the lower bound of θ to 1.1584!
This was undoubtedly a bombshell dropped by AI in the field of pure mathematical discovery.
But the story didn't end there.
Just a week later, on May 22, mathematician Robert Gerbicz, building on AlphaEvolve's discovery, performed a "deep refinement" using traditional human ingenuity.
He cleverly adjusted and extended AlphaEvolve's construction, pushing the parameter θ even higher to 1.173050!
This is no small number; in mathematics, every decimal place of progress can signify a massive leap in theoretical understanding.
Paper link: https://arxiv.org/abs/2505.16105
Terry Tao sincerely marveled at this: "The complementarity of human and AI methods is precisely why mathematics is advancing so rapidly."
AI's "breadth scan" and humanity's "deep refinement" act like dual engines, simultaneously driving mathematical research with long-awaited acceleration.
Then came this latest development, where Terry Tao further revealed an improvement to the parameter θ, increasing it from 1.173050 to 1.173077.
This breakthrough adopted an even more ingenious approach.
The author abandoned AlphaEvolve's computationally intensive brute-force search. Instead, they set the construction parameters from the previous paper to "infinity." Then, by leveraging the principle of "concentration of measure," they directly calculated the limit value of θ.
What's the ingenuity?
Tao explained: "Precisely for this reason, this time it only required a small amount of computer assistance."
The author used a computer program (MATLAB) to carefully try different parameter values and found the maximum value that this formula could yield.
Ultimately, the author proved that the lower bound of θ could be increased to 1.173077.
From "Competition" to "Co-creation"
This pursuit surrounding θ perfectly illustrates the new paradigm of AI and human collaboration in advancing scientific frontiers.
AlphaEvolve can rapidly process vast amounts of information, leveraging its unique algorithms to discover patterns and connections that humans might miss.
The initial constructions it provides are like lighting a lamp in the dark, guiding subsequent exploration.
Mathematicians, on the other hand, bring deep thinking capabilities to the table.
They can understand the essence of AI's discoveries and, based on that, perform abstraction, generalization, and theoretical sublimation. Gerbicz's optimization of parameters and the subsequent breakthrough using asymptotic methods are testaments to the unique value of human intelligence.
In the future, the mutual collaboration between AI and humans will be extremely competitive.
In mathematics, the crowning glory of human intellect, we are witnessing the acceleration of this trend.
In the past, we marveled at AlphaGo defeating human Go grandmasters, which was more of a "competition" relationship.
However, AlphaEvolve's success and the subsequent breakthroughs by mathematicians building on its foundation demonstrate a more anticipated "co-creation" relationship.
AI is no longer just a tool; it is becoming a partner for mathematicians to inspire ideas, broaden perspectives, and accelerate discoveries.
This is not just a victory for mathematics; it also heralds the arrival of a new era of scientific discovery, an era where human intellect and machine intelligence will go hand in hand, jointly exploring the unknown universe.
References:
https://arxiv.org/abs/2506.01896
https://mathstodon.xyz/@tao