LARGE SIGNED SUBSET SUMS. Issue 3 (8th April 2021)

• Record Type: Journal Article
• Authors: Ambrus, Gergely; González Merino, Bernardo
• Is Part Of: Mathematika. Volume 67:Issue 3(2021)
• Page Start: 579
• Page End: 595
• Publication Date: 2021-04-08
• Subjects: 52A37 -- 52A40 -- 05A99 (primary); Mathematics -- Periodicals; 510.5
• DOI: 10.1112/mtk.12091
• Languages: English
• ISSNs: 0025-5793

Abstract

Abstract: We study the following question: for given d ⩾ 2, n ⩾ d and k ⩽ n, what is the largest value c ( d, n, k ) such that from any set of n unit vectors in R d, we may select k vectors with corresponding signs ±1 so that their signed sum has norm at least c ( d, n, k ) ? The problem is dual to classical vector sum minimization and balancing questions, which have been studied for over a century. We give asymptotically sharp estimates for c ( d, n, k ) in the general case. In several special cases, we provide stronger estimates: the quantity c ( d, n, n ) corresponds to the ℓ p ‐polarization problem, while determining c ( d, n, 2 ) is equivalent to estimating the coherence of a vector system, which is a special case of p ‐frame energies. Two new proofs are presented for the classical Welch bound when n = d + 1 . For large values of n, volumetric estimates are applied for obtaining fine estimates on c ( d, n, 2 ) . Studying the planar case, sharp bounds on c ( 2, n, k ) are given. Finally, we determine the exact value of c ( d, d + 1, d + 1 ) under some extra assumptions.