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Displaying similar documents to “A general upper bound in extremal theory of sequences”

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 .

Piatetski-Shapiro sequences via Beatty sequences

Lukas Spiegelhofer (2014)

Acta Arithmetica

Similarity:

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...

On Ozeki’s inequality for power sums

Horst Alzer (2000)

Czechoslovak Mathematical Journal

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Let p ( 0 , 1 ) be a real number and let n 2 be an even integer. We determine the largest value c n ( p ) such that the inequality i = 1 n | a i | p c n ( p ) holds for all real numbers a 1 , ... , a n which are pairwise distinct and satisfy min i j | a i - a j | = 1 . Our theorem completes results of Ozeki, Mitrinović-Kalajdžić, and Russell, who found the optimal value c n ( p ) in the case p > 0 and n odd, and in the case p 1 and n even.

A problem of Rankin on sets without geometric progressions

Melvyn B. Nathanson, Kevin O&#039;Bryant (2015)

Acta Arithmetica

Similarity:

A geometric progression of length k and integer ratio is a set of numbers of the form a , a r , . . . , a r 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 2 i , a 2 i - 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...

On the structure of sequences with forbidden zero-sum subsequences

W. D. Gao, R. Thangadurai (2003)

Colloquium Mathematicae

Similarity:

We study the structure of longest sequences in d which have no zero-sum subsequence of length n (or less). We prove, among other results, that for n = 2 a and d arbitrary, or n = 3 a and d = 3, every sequence of c(n,d)(n-1) elements in d which has no zero-sum subsequence of length n consists of c(n,d) distinct elements each appearing n-1 times, where c ( 2 a , d ) = 2 d and c ( 3 a , 3 ) = 9 .