Linear equations with two variables in Piatetski-Shapiro sequences

Saito, Kota

doi:10.4064/aa201125-26-7

Cited by 3 publications

(3 citation statements)

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“…We do heuristic calculations to construct elements in T (α). In the previous works [Gla17,Sai22], the following two things are important to prove the infinitude of T (PS(α)):…”

Section: Ingredients Of the Proofsmentioning

confidence: 99%

A system of certain linear Diophantine equations on analogs of squares

Kanado¹,

Saito²

2022

Preprint

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Let S(α) be the set of all integers of the form αn 2 for n ≥ α −1/2 , where x denotes the integer part of x. We investigate the existence of tuples (k, , m) of integers such that all of k, , m, k + , + m, m + k, k + + m belong to S(α). Let T (α) be the set of all such tuples. In this article, we reveal that T (α) is infinite for all α ∈ (0, 1) ∩ Q and for almost all α ∈ (0, 1) in the sense of Lebesgue measure. As a corollary, we disclose that the cardinality of α ∈ (0, 1) such that #T (α) = 0 is at most countable. Furthermore, we show that if there exists α > 0 such that T (α) is finite, then there is no perfect Euler brick. We also study the set of all integers of the form αn 2 for n ∈ N.

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“…We do heuristic calculations to construct elements in T (α). In the previous works [Gla17,Sai22], the following two things are important to prove the infinitude of T (PS(α)):…”

Section: Ingredients Of the Proofsmentioning

confidence: 99%

A system of certain linear Diophantine equations on analogs of squares

Kanado¹,

Saito²

2022

Preprint

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“…not solvable) in PS(α). Saito [12] improved Glasscock's result in the case a > b ≥ 0 as follows. Let a = 1 and b be real numbers with a > b ≥ 0.…”

Section: Introductionmentioning

confidence: 96%

On the number of representations of integers as differences between Piatetski-Shapiro numbers

Yoshida

2021

Preprint

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“…For a given set X ⊆ N, we define T (X) as the set of all tuples (k, ℓ, m) ∈ N 3 with k ≤ ℓ ≤ m such that all of (1.1) belong to X. Interestingly, Glasscock found that #T (PS(α)) = ∞ for almost all α ∈ (1, 2) [G17, Corollary 1]. Note that PS(2) = S. The second author improved on this, showing that Glasscock's result remains true even if we replace "for almost all" with "for all"; that is, #T (PS(α)) = ∞ for all α ∈ (1, 2) [S22,Corollary 1.2]. In view of these results, it is natural to consider the following question.…”

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confidence: 99%