On a problem of Erdős and Graham about consecutive sums in strictly increasing sequences

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2024-05-28 DOI:10.1112/blms.13098

Adrian Beker

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引用次数: 0

Abstract

We show the existence of a constant $c > 0$ such that, for all positive integers $n$ , there exist integers $1 \leq a_{1} < \dots < a_{k} \leq n$ such that there are at least $c n^{2}$ distinct integers of the form $\sum_{i = u}^{v} a_{i}$ with $1 \leq u \leq v \leq k$ . This answers a question of Erdős and Graham. We also prove a non-trivial upper bound on the maximum number of distinct integers of this form and address several open problems.

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论厄尔多斯和格雷厄姆关于严格递增序列中连续和的一个问题

我们证明了一个常数 c > 0 $c > 0$ 的存在，即对于所有正整数 n $n$ ，存在整数 1 ≤ a 1 < ⋯ < a k ≤ n $1 \leq a_1 < \cdots < a_k \leq n$ ，这样至少有 c n 2 $cn^2$ 形式为 ∑ i = u v a i $sum _{i=u}^{v}a_i$ 的不同整数，其中 1 ≤ u ≤ v ≤ k $1 \leq u \leq v \leq k$ 。这回答了厄尔多斯和格雷厄姆的一个问题。我们还证明了关于这种形式的不同整数的最大数目的非难上限，并解决了几个悬而未决的问题。

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