Infinite Sequences with Finite Cross-Correlation

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.