For every integer d≥ 2, every prime p, every H∈[1,p) satisfying ln ln p=o(ln H), and every coefficient vector a=(a_1,…,a_d)∈( Z/p Z)^d, one has displaystyle min_1≤ n≤ H n∏_i=1^d|fraca_i np|asymp frac1(ln p)^d-1ln H, with absolute implied constants.

Littlewood’s discrete conjecture

For every integer $d\ge 2$ , every prime $p$ , every $H\in[1,p)$ satisfying $\ln\ln p=o(\ln H)$ , and every coefficient vector $a=(a_1,\ldots,a_d)\in(\mathbb Z/p\mathbb Z)^d$ , one has $\displaystyle \min_{1\le n\le H} n\prod_{i=1}^d\left\|\frac{a_i n}{p}\right\|\asymp\frac{1}{(\ln p)^{d-1}\ln H}$ , with absolute implied constants.

References

Primary source

arXiv

Additional references

On the cardinality of the exception set in Littlewood's discrete conjecture — arXiv

additive-combinatorics analytic-number-theory

Posted 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed progress

A new unrefereed preprint claims the expected logarithmic behavior for almost every allowed coefficient choice, but the conjecture for every choice remains open.

Littlewood’s discrete conjecture predicts logarithmic-order behavior for all coefficient vectors under the stated growth condition. The new result addresses only a density-one family, not the universal conjecture.

September 8, 2026 almost-all theorem

An arXiv preprint claims the predicted logarithmic order for almost all coefficient vectors satisfying the growth condition. This is substantial progress toward the conjectural scale, but it does not settle the exceptional vectors or the full conjecture.

Current status (as of September 2026): The predicted scale is claimed for almost all admissible vectors, while the assertion for every vector remains open and the new result is unverified.

Sources

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