At 10^81 Objects, the Imbalance Is Still About Three

Here is a balancing problem that gets nasty very quickly.

Suppose you have to divide a large group of volunteers into two teams. You care about more than head count. You also want roughly equal totals for first-aid experience, hours available, languages spoken, equipment skills, and perhaps dozens of other things. Moving one person to fix the language imbalance can make the medical imbalance worse. Add enough people and enough categories and a tidy split stops looking obvious.

Discrepancy theory studies mathematical versions of this problem. A famous one, the Komlós problem, asks how well a collection of many-dimensional objects can always be divided into two sides when no single object is too large.

For nearly 30 years, the best general guarantee grew with the square root of the logarithm of the number of objects. Nikhil Bansal and Haotian Jiang have now replaced that with something much smaller: roughly the fourth root of the logarithm.

That phrase is not designed to produce excitement in ordinary people. The number is better.

Quanta Magazine gives an intentionally absurd example. Make the problem contain about 10⁸¹ objects, roughly as many as the estimated number of atoms in the observable universe. The new size-dependent term is still only around three under the article's normalization.

The Komlós conjecture says the answer should ultimately be bounded by a constant. Three sounds suspiciously like a constant. It isn't, quite.

What is actually being balanced?

The formal version starts with columns of numbers. Think of each column as one object and each row as one property we want to balance. Every object gets a sign, +1 or -1, corresponding to which of two groups receives it. Add the signed columns together and look at every row. The discrepancy is the largest imbalance that remains.

There is one important restriction: each column has Euclidean length at most 1. In the volunteer analogy, no one person is allowed to carry an arbitrarily huge amount of the quantities being balanced. Real scheduling problems do not automatically satisfy this assumption; the analogy is only there to make the structure visible.

The Komlós conjecture, proposed in the early 1980s, says there should be some universal ceiling on the discrepancy no matter how many columns the problem contains. Mathematicians still do not know whether that is true.

In 1998, Wojciech Banaszczyk proved a celebrated bound that, in this setting, leads to growth on the order of √log n. That is already slow. Bansal and Jiang's new result pushes the dependence down to roughly (log n)^1/4, with additional very slowly growing factors hidden in the formal theorem.

A fourth root is a square root taken twice. The logarithm has already compressed an enormous number before the square roots even begin.

This is why 10⁸¹ is useful. The example makes a distinction that notation hides: a function can increase forever while increasing so slowly that ordinary physical scale barely moves it.

The title's “about three” needs one warning label. Big-O notation suppresses multiplicative constants, the formal result includes poly(log log n) factors, and the exact numerical value depends on normalization. Bansal and Jiang have not proved that every literal 10⁸¹-object allocation can be balanced to exactly three units. The point is the growth rate. The part of the guarantee that depends on the sheer number of objects has become astonishingly reluctant to grow.

The proof also gives an algorithm

This is not only an existence result. Bansal and Jiang give polynomial-time algorithms for finding the colorings, or +1/-1 assignments, behind their bounds.

The technical machinery involves a discrete Brownian motion guided by a semidefinite program. The new ingredient is what the authors call affine spectral independence. At a high level, it helps control dependencies among the different constraints so that too many rows do not drift into large imbalance together.

That method may prove as interesting as the exponent. A 2026 paper by Haotian Jiang and other researchers has already used the same general framework to obtain improved constructive results for related prefix-discrepancy and Steinitz problems.

There is still a large conceptual gap between “grows absurdly slowly” and “does not grow at all.” The Komlós conjecture wants the second one. A proof that the discrepancy is bounded by a true universal constant would finish a problem that has resisted mathematicians for decades.

Bansal and Jiang have instead given us an odd intermediate state. The known bound still goes to infinity. You may need more objects than the universe has atoms before it begins to feel much like it.

Sources

• Nikhil Bansal and Haotian Jiang, “Decoupling via Affine Spectral-Independence: Beck-Fiala and Komlós Bounds Beyond Banaszczyk”
• Max G. Levy, Quanta Magazine, “‘Huge Breakthrough’ in the Math of Imbalance” (August 21, 2026)
• Kunal Dutta, Agastya Vibhuti Jha, and Haotian Jiang, “Near-Optimal Constructive Bounds for ℓ2 Prefix Discrepancy and Steinitz Problems via Affine Spectral Independence” (2026)