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Infinite Sequences with Finite Cross-Correlation

  • Solomon W. Golomb
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6338)

Abstract

Let \(A = \{a_k\}^\infty_{k = 1}\) be an infinite increasing sequence of positive integers. We define the infinite binary sequence \(\overline{A} = \{\alpha_j\}_{j=1}^\infty\) to have α j  = 1 if j ∈ A, α j  = 0 if j ∉ A (including when j ≤ 0). If \(B = \{b_k\}_{k=1}^\infty\) is also an infinite increasing sequence of positive integers with \(\overline{B} = \{\beta_j\}_{j = 1}^\infty\), by the “cross-correlation of A and B” we will mean the un-normalized, infinite-domain cross-correlation of \(\overline{A}\) and \(\overline{B}\), i.e.
$$ C_{AB}(\tau) = \sum \limits_{i = 1}^\infty \alpha_i\beta_{i + \tau} $$
for all τ ∈ Z.

Our interest will be in identifying pairs of sequences A and B for which C AB (τ) is finite for all τ ∈ Z, and especially when C AB (τ) < K for some uniform bound K, for all τ ∈ Z. We will exhibit pairs of sequences A and B where C AB (τ) ≤ 1 for all τ ∈ Z. If B = P = {p 1, p 2, p 3, ...} = {2, 3, 5, 7,...} is the sequence of the prime numbers, we will exhibit sequences A such that C AP (τ) is finite for all τ ∈ Z, and question whether a sequence A exists such that C AP (τ) < K for some uniform bound K and all τ ∈ Z.

Keywords

Positive Integer Prime Number Greedy Algorithm Binary Sequence Integer Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Problem 10208. Amer. Math. Monthly (1992)Google Scholar
  2. 2.
    Golomb, S.W., Taylor, H.: Cyclic projective planes, perfect circular rulers, and good spanning rulers. In: Sequences and their Applications, Bergen, pp. 166–181 (2001); Discrete Math. Theor. Comput. Sci. (Lond.), Springer, London (2002) Google Scholar
  3. 3.
    Ribenboim, P.: The little book of bigger primes, 2nd edn. Springer, New York (2004)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Solomon W. Golomb
    • 1
  1. 1.University of Southern CaliforniaLos AngelesUSA

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