2017
DOI: 10.4064/aa8355-10-2016
|View full text |Cite
|
Sign up to set email alerts
|

Solutions to certain linear equations in Piatetski-Shapiro sequences

Abstract: Denote by PS(α) the image of the Piatetski-Shapiro sequence n → ⌊n α ⌋, where α > 1 is non-integral and ⌊x⌋ is the integer part of x ∈ R. We partially answer the question of which bivariate linear equations have infinitely many solutions in PS(α): if a, b ∈ R are such that the equation y = ax + b has infinitely many solutions in the positive integers, then for Lebesgue-a.e. α > 1, it has infinitely many or at most finitely many solutions in PS(α) according as α < 2 (and 0 ≤ b < a) or α > 2 (and (a, b) = (1, 0)… Show more

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
1
1
1

Citation Types

1
13
0

Year Published

2020
2020
2022
2022

Publication Types

Select...
4
1

Relationship

1
4

Authors

Journals

citations
Cited by 8 publications
(14 citation statements)
references
References 11 publications
1
13
0
Order By: Relevance
“…To this end, fix j 1 < θ 1 < θ 2 < j 2 . In what follows, the phrase "for all sufficiently large n" means "for all n ≥ n 0 ," where n 0 ∈ N may depend on any of the quantities and sequences introduced so far, including θ 1 and θ 2 .…”
Section: Proof Of Main Theoremmentioning
confidence: 99%
“…To this end, fix j 1 < θ 1 < θ 2 < j 2 . In what follows, the phrase "for all sufficiently large n" means "for all n ≥ n 0 ," where n 0 ∈ N may depend on any of the quantities and sequences introduced so far, including θ 1 and θ 2 .…”
Section: Proof Of Main Theoremmentioning
confidence: 99%
“…We have investigated distributions of APs contained in a Piatetski-Shapiro sequence. Since it is known that the Piatetski-Shapiro sequence with exponent α ∈ (1, 2) contains mathematical structures other than APs, e.g., solutions to x + y = z [18], it might be also interesting to investigate the distribution of such solutions. As other natural questions, we have the positive-density version and prime-number version.…”
Section: Future Workmentioning
confidence: 99%
“…[18] mentioned that for every α ∈ (1, 2), the equation x + y = z has infinitely many solutions in PS(α). Glasscock [18] showed that if the equation y = ax + b with real a > b ≥ 0 has infinitely many solutions in N, then, for Lebesgue-a.e. α ∈ (1, 2), the equation y = ax + b has infinitely many solutions in PS(α).…”
Section: Introductionmentioning
confidence: 99%
“…In this article, we discuss the solvability in PS(α) of the equation (1.1) y = ax + b for fixed a, b ∈ R with a / ∈ {0, 1}. Glasscock proved that if equation (1.1) is solvable in N, then for Lebesgue almost every α > 1, it is solvable or not in PS(α) according as α < 2 or α > 2 [Gla17,Gla20]. In addition, as a corollary, he showed that for Lebesgue almost every 1 < α < 2, there are infinitely many (k, ℓ, m) ∈ N 3 such that…”
mentioning
confidence: 99%