O(sqrt d) bound conjecture for the signed series and Steinitz problems in the ℓ2 norm

Establish that the optimal bounds in dimension d for both (i) the signed series problem in the ℓ2 norm—given vectors v1,…,vn in R^d with ℓ2-norm at most 1, determine signs x(i) in {−1,1} such that max_{i∈[n]} ||∑_{j=1}^i x(j) v_j||_2 = O(√d)—and (ii) the Steinitz problem in the ℓ2 norm—given zero-sum vectors v1,…,vn in R^d with ℓ2-norm at most 1, find a permutation π with sup_{k≤n} ||∑_{i=1}^k v_{π(i)}||_2 = O(√d).

Background

The paper studies the signed series (prefix discrepancy) problem and its close relation to the Steinitz problem. Both concern bounding partial sums of vectors in Rd with unit ℓ2-norm, and both are conjectured to admit bounds scaling as O(√d).

Banaszczyk (2012) proved a non-constructive O(√d + √log n) bound for both problems, which is the best known towards the conjecture. Prior constructive bounds were O(√(d log n)); this paper gives a randomized polynomial-time algorithm achieving O(√d + log n), which is near the conjectured O(√d) target but still leaves the conjecture open.

References

It has been conjectured that the actual bound for both the Steinitz and the signed series problems is O{\sqrt{d}.

— Constructive l2-Discrepancy Minimization with Additive Deviations  (2508.21423 - Dutta, 29 Aug 2025) in Section 1, Introduction

We do not know whether our approach extends to the strong Komlós conjecture.

— An elementary proof of the Komlós conjecture  (2609.20979 - Karingula et al., 17 Sep 2026) in Introduction, immediately following the proof of Theorem 1 (Theorem \ref{thm:komlos})