DeMath — Decentralized math research infrastructure

Statement

Tao (2015) proved Erdős's discrepancy conjecture: for every sequence

f : \mathbb{N} \to \{-1, +1\}

,

\sup_{N, d \in \mathbb{N}} \left| \sum_{j=1}^{N} f(jd) \right| = \infty

. The proof is non-effective. Define

D(N) := \min_{f} \max_{n \leq N,\, d \geq 1} \left| \sum_{j=1}^{n} f(jd) \right|

where the minimum runs over all

\pm 1

sequences. \textbf{Task:} Establish an explicit quantitative growth rate for

D(N) \to \infty

, i.e., an unconditional lower bound of the form

D(N) \geq g(N)

for an explicit unbounded function

g

.

Current frontier

Tao (2015) qualitatively proved the discrepancy supremum is unbounded but gave no explicit rate; SAT-derived discrete lower bound D(N) ≥ 3 for N ≥ 1161 (Polymath 5, 2014).

When this counts as solved

Tao's 2015 proof showed the discrepancy supremum is unbounded but gave no explicit rate. Closing this means establishing matching unconditional upper and lower bounds for

D(N)

up to constants — the asymptotic growth rate is pinned down. A strict advance also counts: any explicit unconditional lower bound

D(N) \geq g(N)

for an unbounded

g

(e.g.,

\log \log N

,

(\log \log N)^c

,

\log \log \log N

, anything stronger) — the literature has no such effective bound, so any explicit

g

qualifies. An upper bound stronger than the trivial

D(N) = O(\sqrt{N})

also counts. A counterexample doesn't apply. New exact values of

D(N)

for small

N

via SAT, or refinements of Tao's argument that don't yield an explicit rate, don't close the problem.

Classification

quantitative