On the distribution of sums of vectors in general position

An analogue of the Littlewood-Offord problem posed by the first author is to find the maximum number of subset sums equal to the same vector over all ways of selecting n vectors in R in general position. This note reviews past progress and motivation for this problem, and presents a construction that gives a respectable new lower bound, Ω(2nn1−3d/2), which compares for d ≥ 2 to the previously known upper bound O(2nn−1−d/2). Running head: Sums of Vectors in General Position 1 Research supported in part by NSF grant DMS–9701211. 2 Research supported in part by the Spezialforschungbereich “Optimierung und Kontrolle” Sums of Vectors in General Position One version of the famous Littlewood-Offord problem [11] asks how to select complex numbers a1, . . . , an, not necessarily distinct, with each |ai| ≥ 1, and a target open ball T ⊆ C of unit diameter to maximize the number of the 2 subset sums ∑ i∈I ai, where I ⊆ [n], lying in T . Viewing C as R, one can extend this problem to arbitrary dimension d, and ask the same thing, where now the ai’s are vectors in R. By setting all ai equal to the same vector, it is possible to have ( n b2 c ) subset sums lying inside T . Erdős [3] showed this was best-possible for the reals (d = 1); Katona [8] and Kleitman [9] independently proved the same for the original case of complex numbers (d = 2); Kleitman [10] later found an ingenious inductive proof that ( n b2 c ) is best-possible for general d. Even if we restrict the target set to just a single point t, this bound is still achieved. But what if we must also spread out the vectors ai in the sense of asking that any d of them be linearly independent? Had we only needed to hit a unit diameter ball target, the answer would have remained at ( n b2 c ) , but by shrinking the target to a single point, it will be tougher in general to get as many sums to hit the target. With a single point target, the restriction that each |ai| ≥ 1 no longer affects the answer, so it can be dropped. Therefore, we are now interested in the following: General Position Subset Sum Problem. Given positive integers n, d, how can one select vectors a1, . . . , an ∈ R and a target t ∈ R to achieve the maximum number fd(n) of the 2 n subset sums ∑ i∈I ai, where I ⊆ [n], equal to t, provided that every d of the vectors ai are linearly independent? Griggs [5] arrived at exactly this problem in connection with a model of database security. In the database security studies of Mirka Miller et al. [13,12,1, cf. 4], there is a database of numerical records, {x1, . . . , xn}, e.g., the salaries of the n members of a department. One may request the sums ∑ j∈J xj of certain subsets J ⊆ {1, . . . , n}, and an answer will be given by the control mechanism, provided that no “compromise” results. In the basic model, a compromise means that the requester is able to determine, by taking an appropriate linear combination of the answered queries (sums), some individual entry xi. The problem is to maximize the number of queries that can be answered without compromising the database. Griggs proposed an extension of this problem to prevent compromise by anyone with prior knowledge of d− 1 records: We say a compromise results whenever one can determine some linear combination of at most d records, ∑ j∈J αjxj , where all αj 6= 0 and 0 < |J | ≤ d. It turns out that the maximum number of queries that can be answered without compromise is precisely fd(n). In one dimension, f1(n) is equivalent to the real case of the Littlewood-Offord problem, and so f1(n) = ( n ⌊ n 2 ⌋ ) . Determining fd(n) for fixed dimension d ≥ 2 remains an intriguing and funda-