Balanced Subset Sums in Dense Sets of Integers
نویسنده
چکیده
Let 1 ≤ a1 < a2 < · · · < an ≤ 2n − 2 denote integers. Assuming that n is large enough, we prove that there exist ε1, . . . , εn ∈ {−1,+1} such that |ε1 + · · ·+εn| ≤ 1 and |ε1a1+ · · ·+εnan| ≤ 1. This result is sharp, and in turn it confirms a conjecture of Lev. We also prove that when n is even, every integer in a large interval centered at (a1 + a2 + · · · + an)/2 can be represented as the sum of n/2 elements of the sequence.
منابع مشابه
Probability, Networks and Algorithms Probability, Networks and Algorithms Balanced subset sums of dense sets of integers
Let 1 ≤ a1 < a2 < . . . < an ≤ 2n − 2 denote integers. We prove that there exist ε1, . . . , εn ∈ {−1,+1} such that |ε1 + . . . + εn| ≤ 1 and |ε1a1+ . . .+ εnan| ≤ 1, at least when n is large enough. This result is sharp and, in turn, confirms a conjecture of V.F. Lev. We also prove that more than n/12 consecutive integers can be reperesented as the sum of roughly n/2 elements of the sequence.
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