On a sumset problem of dilates

Chahal, Sandeep Singh; Pandey, Ram Krishna

doi:10.1007/s13226-021-00089-6

Cited by 3 publications

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“…In 2013, Balog et al [1] proved that |p • A + q • A| ≥ (p + q)|A| − (pq) (p+q−3)(p+q)+1 , where p < q are relatively primes and A ⊆ Z. In 2020, Chahal and Pandey [4] handled the case for the cardinality of 3 • A + k • A, under some conditions on A and also generalized this result for q • A + k • A, where q < k is an odd prime. In 2017, Freiman et al [8] proved that if r ≥ 3, then |A + r • A| ≥ 4|A| − 4.…”

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On direct and inverse problems related to some dilated sumsets

Kaur,

Singh

2024

Comptes Rendus. Mathématique

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Let A be a nonempty finite set of integers. For a real number m, the set m • A = {ma : a ∈ A} denotes the set of m-dilates of A. In 2008, Bukh initiated an interesting problem of finding a lower bound for the sumset of dilated sets, i.e., a lower bound forwhere λ 1 , λ 2 , . . . , λ h are integers and A be a subset of integers. In particular, for nonempty finite subsets A and B , the problem of dilates of A and B is defined as A + k • B = {a + kb : a ∈ A and b ∈ B }. In this article, we obtain the lower bound for the cardinality of A + k • B with k ≥ 3 and describe sets for which equality holds. We also derive an extended inverse result with some conditions for the sumset A + 3 • B .

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confidence: 99%

On direct and inverse problems related to some dilated sumsets

Kaur,

Singh

2024

Comptes Rendus. Mathématique

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On Sumset Problems and Their Various Types

Kaur¹,

Singh²

2022

Trends in Mathematics

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On a sumset problem for dilated integer sets

Kaur,

Bhanja,

Singh

2024

Int. J. Number Theory

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Let [Formula: see text] be a non-empty finite set of integers. For any integer [Formula: see text], let [Formula: see text]. One of the important problems in additive combinatorics is to find the optimal lower bound for the cardinality of the sumset [Formula: see text]. Cilleruelo, Silva, and Vinuesa conjectured that, if [Formula: see text] is a positive integer and [Formula: see text] is a finite set of integers with sufficiently large cardinality, then [Formula: see text]. For some values of [Formula: see text] this conjecture is already confirmed. In this article, under certain restrictions on set [Formula: see text], we confirm this conjecture for all positive integers [Formula: see text] except for those of the form [Formula: see text] or [Formula: see text] where [Formula: see text] is an odd prime number and [Formula: see text] for [Formula: see text].

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On a sumset problem of dilates

Cited by 3 publications

References 7 publications

On direct and inverse problems related to some dilated sumsets

On direct and inverse problems related to some dilated sumsets

On Sumset Problems and Their Various Types

On a sumset problem for dilated integer sets

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