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0 - 1 sequences having the same numbers of ( 1 - 1 ) -couples of given distances

Antonín Lešanovský, Jan Rataj, Stanislav Hojek (1992)

Mathematica Bohemica

Let a be a 0 - 1 sequence with a finite number of terms equal to 1. The distance sequence δ ( a ) of a is defined as a sequence of the numbers of ( 1 - 1 ) -couples of given distances. The paper investigates such pairs of 0 - 1 sequences a , b that a is different from b and δ ( a ) = δ ( b ) .

A basis of ℤₘ, II

Min Tang, Yong-Gao Chen (2007)

Colloquium Mathematicae

Given a set A ⊂ ℕ let σ A ( n ) denote the number of ordered pairs (a,a’) ∈ A × A such that a + a’ = n. Erdős and Turán conjectured that for any asymptotic basis A of ℕ, σ A ( n ) is unbounded. We show that the analogue of the Erdős-Turán conjecture does not hold in the abelian group (ℤₘ,+), namely, for any natural number m, there exists a set A ⊆ ℤₘ such that A + A = ℤₘ and σ A ( n ̅ ) 5120 for all n̅ ∈ ℤₘ.

A basis of Zₘ

Min Tang, Yong-Gao Chen (2006)

Colloquium Mathematicae

Let σ A ( n ) = | ( a , a ' ) A ² : a + a ' = n | , where n ∈ N and A is a subset of N. Erdős and Turán conjectured that for any basis A of order 2 of N, σ A ( n ) is unbounded. In 1990, Imre Z. Ruzsa constructed a basis A of order 2 of N for which σ A ( n ) is bounded in the square mean. In this paper, we show that there exists a positive integer m₀ such that, for any integer m ≥ m₀, we have a set A ⊂ Zₘ such that A + A = Zₘ and σ A ( n ̅ ) 768 for all n̅ ∈ Zₘ.

A characterization of sequences with the minimum number of k-sums modulo k

Xingwu Xia, Yongke Qu, Guoyou Qian (2014)

Colloquium Mathematicae

Let G be an additive abelian group of order k, and S be a sequence over G of length k+r, where 1 ≤ r ≤ k-1. We call the sum of k terms of S a k-sum. We show that if 0 is not a k-sum, then the number of k-sums is at least r+2 except for S containing only two distinct elements, in which case the number of k-sums equals r+1. This result improves the Bollobás-Leader theorem, which states that there are at least r+1 k-sums if 0 is not a k-sum.

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