Ford–Green–Koukoulopoulos close-divisors and equal-subset-sums conjecture

For each fixed integer $k\ge 2$ , define $\alpha_k$ as the supremum of the real numbers $a$ such that, for almost every positive integer $n$ , there exist $k$ distinct divisors of $n$ contained in a multiplicative interval of relative length $(\log n)^{-a}$ . Independently include each positive integer $i$ in a random set $\mathbf A$ with probability $1/i$ , and define $\beta_k$ as the supremum of the real numbers $c<1$ such that, as $D\to\infty$ with probability tending to $1$ , the set $\mathbf A\cap[D^c,D]$ contains $k$ distinct subsets having equal sums. The conjecture is that $\alpha_k=\frac{\beta_k}{1-\beta_k}$ for every fixed $k\ge 2$ .

References

Primary source

arXiv

Additional references

Equal subset sums and close divisors — arXiv — Jianfeng Hou, Hongbin Zhao

additive-combinatorics analytic-number-theory

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

A September 2026 preprint claims to prove the conjecture, but the result has not been independently verified.

Ford, Green, and Koukoulopoulos conjectured an exact relation between close-divisor and repeated-subset-sum concentration exponents, namely $\alpha_k=\beta_k/(1-\beta_k)$ for fixed $k\ge 2$ . Their framework appeared in 2019 and was published in 2023.

Known results

Ford, Green, and Koukoulopoulos established the lower bound $\alpha_k\ge\beta_k/(1-\beta_k)$ and presented equality as a conjecture; their work also characterized $\beta_k$ through an optimization problem.

September 2026 claimed proof

On September 22, 2026, Jianfeng Hou and Hongbin Zhao’s preprint Equal subset sums and close divisors reported the equality $\alpha_k=\beta_k/(1-\beta_k)$ , using flag refinement, entropy concavity, and a uniform approximate-subset-sum bound. A separate September 2026 preprint likewise claims the converse inequality and the conjecture’s proof, together with an entropy-threshold identity.

Current status (as of September 2026): The conjectured equality is claimed in recent arXiv preprints, but remains unverified; the earlier lower bound is established.

Sources

No solutions have been posted yet.