EXTENDED INVERSE THEOREMS FOR RESTRICTED SUMSET IN INTEGERS

Article Type

Research Article

Publication Title

Bulletin of the Korean Mathematical Society

Abstract

Let h and k be positive integers such that h ≤ k. Let A = {a0, a1, …, ak−1 } be a nonempty finite set of k integers. The h-fold sumset, denoted by hA, is a set of integers that can be expressed as a sum of h elements (not necessarily distinct) of A. The restricted h-fold sumset, denoted by h∧ A, is a set of integers that can be expressed as a sum of h distinct elements of A. The characterization of the underlying set for small deviation from the minimum size of the sumset is called an extended inverse problem. Freiman studied such a problem and proved a theorem for 2A, which is known as Freiman’s 3k − 4 theorem. Very recently, Tang and Xing, and Mohan and Pandey studied some more extended inverse problems for the sumset hA, where h ≥ 2. In this article, we prove some extended inverse theorems for sumsets 2∧ A, 3∧ A and 4∧ A. In particular, we classify the set(s) A for which |2∧ A| = 2k − 2, |2∧ A| = 2k − 1, and |2∧ A| = 2k. Furthermore, we also classify set(s) A when |3∧ A| = 3k − 7, |3∧ A| = 3k − 6, and |4∧ A| = 4k − 14.

First Page

1339

Last Page

1367

DOI

10.4134/BKMS.b230631

Publication Date

9-1-2024

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