Wyniki wyszukiwania

1

The sequence of fractional parts of roots

100%

O'Bryant K.

Acta Arithmetica

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2015

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tom 169

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nr 4

357-371

EN

We study the function $M_{θ}(n) = ⌊1/{θ^{1/n}}⌋$, where θ is a positive real number, ⌊·⌋ and {·} are the floor and fractional part functions, respectively. Nathanson proved, among other properties of $M_{θ}$, that if log θ is rational, then for all but finitely many positive integers n, $M_{θ}(n) = ⌊n/log θ - 1/2⌋$. We extend this by showing that, without any condition on θ, all but a zero-density set of integers n satisfy $M_{θ}(n) = ⌊n/log θ - 1/2⌋$. Using a metric result of Schmidt, we show that almost all θ have asymptotically (log θ log x)/12 exceptional n ≤ x. Using continued fractions, we produce uncountably many θ that have only finitely many exceptional n, and also give uncountably many explicit θ that have infinitely many exceptional n.

2

A discrete Fourier kernel and Fraenkel's tiling conjecture

63%

Graham R., O'Bryant K.

Acta Arithmetica

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2005

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tom 118

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nr 3

283-304

3

A problem of Rankin on sets without geometric progressions

63%

Nathanson M., O'Bryant K.

Acta Arithmetica

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2015

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tom 170

|

nr 4

327-342

EN

A geometric progression of length k and integer ratio is a set of numbers of the form ${a,ar,...,ar^{k-1}}$ for some positive real number a and integer r ≥ 2. For each integer k ≥ 3, a greedy algorithm is used to construct a strictly decreasing sequence $(a_i)_{i=1}^{∞}$ of positive real numbers with a₁ = 1 such that the set $G^{(k)} = ⋃ _{i=1}^{∞} (a_{2i}, a_{2i-1}]$ contains no geometric progression of length k and integer ratio. Moreover, $G^{(k)}$ is a maximal subset of (0,1] that contains no geometric progression of length k and integer ratio. It is also proved that there is a strictly increasing sequence $(A_i)_{i=1}^{∞}$ of positive integers with A₁ = 1 such that $a_i = 1/A_i$ for all i = 1,2,.... The set $G^{(k)}$ gives a new lower bound for the maximum cardinality of a subset of {1,...,n} that contains no geometric progression of length k and integer ratio.

4

Binary linear forms over finite sets of integers

45%

Nathanson M., O'Bryant K., Orosz B., Ruzsa I., Silva M.

Acta Arithmetica

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2007

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tom 129

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nr 4

341-361

Ograniczanie wyników

4 Acta Arithmetica

4 O'Bryant K.

2 Nathanson M.

1 Graham R.

1 Orosz B.

1 Ruzsa I.

1 Silva M.

2 2015

1 2007

1 2005

The sequence of fractional parts of roots

A discrete Fourier kernel and Fraenkel's tiling conjecture

A problem of Rankin on sets without geometric progressions

Binary linear forms over finite sets of integers