منابع مشابه
On generalized Stanley sequences
Let N denote the set of all nonnegative integers. Let k ≥ 3 be an integer and A0 = {a1, . . . , at} (a1 < . . . < at) be a nonnegative set which does not contain an arithmetic progression of length k. We denote A = {a1, a2, . . . } defined by the following greedy algorithm: if l ≥ t and a1, . . . , al have already been defined, then al+1 is the smallest integer a > al such that {a1, . . . , al}...
متن کاملNovel structures in Stanley sequences
Given a set of integers with no three in arithmetic progression, we construct a Stanley sequence by adding integers greedily so that no arithmetic progression is formed. This paper offers two main contributions to the theory of Stanley sequences. First, we characterize well-structured Stanley sequences as solutions to constraints in modular arithmetic, defining the modular Stanley sequences. Se...
متن کاملOn the growth of Stanley sequences
A set is said to be 3-free if no three elements form an arithmetic progression. Given a 3-free set A of integers 0 = a0 < a1 < · · · < at, the Stanley sequence S(A) = {an} is defined using the greedy algorithm: For each successive n > t, we pick the smallest possible an so that {a0, a1, . . . , an} is 3-free and increasing. Work by Odlyzko and Stanley indicates that Stanley sequences may be div...
متن کاملShort Character Sums with Beatty Sequences
Abstract. We estimate multiplicative character sums taken on the values of a nonhomogeneous Beatty sequence {⌊αn+ β⌋ : n = 1, 2, . . . }, where α, β ∈ R, and α is irrational. In particular, our bounds imply that for fixed α, β and a small real number ε > 0, if p is sufficiently large and p1/3+ε ≤ N ≤ p1/2+ε, then among the first N elements of the Beatty sequence there are N/2+ o(N) quadratic no...
متن کاملBarker sequences of odd length
A Barker sequence is a binary sequence for which all non-trivial aperiodic autocorrelations are at most 1 in magnitude. An old conjecture due to Turyn asserts that there is no Barker sequence of length greater than 13. In 1961, Turyn and Storer gave an elementary, though somewhat complicated, proof that this conjecture holds for odd lengths. We give a new and simpler proof of this result.
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ژورنال
عنوان ژورنال: Journal of Combinatorics
سال: 2018
ISSN: 2156-3527,2150-959X
DOI: 10.4310/joc.2018.v9.n4.a2