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Displaying similar documents to “Measure of noncompactness of linear operators between spaces of sequences that are ( N ¯ , q ) summable or bounded”

Sum and difference sets containing integer powers

Quan-Hui Yang, Jian-Dong Wu (2012)

Czechoslovak Mathematical Journal

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Let n > m 2 be positive integers and n = ( m + 1 ) + r , where 0 r m . Let C be a subset of { 0 , 1 , , n } . We prove that if | C | > n / 2 + 1 if m is odd , m / 2 + δ if m is even , where x denotes the largest integer less than or equal to x and δ denotes the cardinality of even numbers in the interval [ 0 , min { r , m - 2 } ] , then C - C contains a power of m . We also show that these lower bounds are best possible.

On Ordinary and Standard Lebesgue Measures on

Gogi Pantsulaia (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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New concepts of Lebesgue measure on are proposed and some of their realizations in the ZFC theory are given. Also, it is shown that Baker’s both measures [1], [2], Mankiewicz and Preiss-Tišer generators [6] and the measure of [4] are not α-standard Lebesgue measures on for α = (1,1,...).

Interpolating sequences, Carleson measures and Wirtinger inequality

Eric Amar (2008)

Annales Polonici Mathematici

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Let S be a sequence of points in the unit ball of ℂⁿ which is separated for the hyperbolic distance and contained in the zero set of a Nevanlinna function. We prove that the associated measure μ S : = a S ( 1 - | a | ² ) δ a is bounded, by use of the Wirtinger inequality. Conversely, if X is an analytic subset of such that any δ -separated sequence S has its associated measure μ S bounded by C/δⁿ, then X is the zero set of a function in the Nevanlinna class of . As an easy consequence, we prove that if S is a dual...

Piatetski-Shapiro sequences via Beatty sequences

Lukas Spiegelhofer (2014)

Acta Arithmetica

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Integer sequences of the form n c , where 1 < c < 2, can be locally approximated by sequences of the form ⌊nα+β⌋ in a very good way. Following this approach, we are led to an estimate of the difference n x φ ( n c ) - 1 / c n x c φ ( n ) n 1 / c - 1 , which measures the deviation of the mean value of φ on the subsequence n c from the expected value, by an expression involving exponential sums. As an application we prove that for 1 < c ≤ 1.42 the subsequence of the Thue-Morse sequence indexed by n c attains both of its values with...

The postage stamp problem and arithmetic in base r

Amitabha Tripathi (2008)

Czechoslovak Mathematical Journal

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Let h , k be fixed positive integers, and let A be any set of positive integers. Let h A : = { a 1 + a 2 + + a r : a i A , r h } denote the set of all integers representable as a sum of no more than h elements of A , and let n ( h , A ) denote the largest integer n such that { 1 , 2 , ... , n } h A . Let n ( h , k ) : = max A : n ( h , A ) , where the maximum is taken over all sets A with k elements. We determine n ( h , A ) when the elements of A are in geometric progression. In particular, this results in the evaluation of n ( h , 2 ) and yields surprisingly sharp lower bounds for n ( h , k ) , particularly for k = 3 .

Finite canonization

Saharon Shelah (1996)

Commentationes Mathematicae Universitatis Carolinae

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The canonization theorem says that for given m , n for some m * (the first one is called E R ( n ; m ) ) we have for every function f with domain [ 1 , , m * ] n , for some A [ 1 , , m * ] m , the question of when the equality f ( i 1 , , i n ) = f ( j 1 , , j n ) (where i 1 < < i n and j 1 < j n are from A ) holds has the simplest answer: for some v { 1 , , n } the equality holds iff v i = j . We improve the bound on E R ( n , m ) so that fixing n the number of exponentiation needed to calculate E R ( n , m ) is best possible.