# Infinite Sequences with Finite Cross-Correlation

Conference paper

## 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 for all

*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.$$ C_{AB}(\tau) = \sum \limits_{i = 1}^\infty \alpha_i\beta_{i + \tau} $$

*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
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## References

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