Testing periodicity

We study the string-property of being periodic and having periodicity smaller than a given bound. Let ∑ be a fixed alphabet and let p,n be integers such that p ≤ n/2. A length-n string over ∑, α=(α ₁,⋯,α _n ), has the property Period(p) if for every i,j {1,⋯,n}, α _i =α _j whenever ij (mod∈p). For an integer parameter g ≤ n/2, the property Period(≥g) is the property of all strings that are in Period(p) for some p≥g. The property ≤ n/2) is also called Periodicity. An ε-test for a property P of length-n strings is a randomized algorithm that for an input α distinguishes between the case that α is in P and the case where one needs to change at least an ε-fraction of the letters of α to get a string in P. The query complexity of the ε-test is the number of letter queries it makes for the worst case input string of length n. We study the query complexity of ε-tests for Period(≥g) as a function of the parameter g, when g varies from 1 to n/2, while ignoring the exact dependence on the proximity parameter ε. We show that there exists an exponential phase transition in the query complexity around g=log∈n. That is, for every δ>0 and g≥(log∈n) ^1+δ, every two-sided error, adaptive ε-test for Period(≥g) has a query complexity that is polynomial in g. On the other hand, for g ≤ log n/6, there exists a one-sided error, non-adaptive ε-test for Period(≥g), whose query complexity is poly-logarithmic in g. We also prove that the asymptotic query complexity of one-sided error non-adaptive ε-tests for Periodicity is Θ(√n log n), ignoring the dependence on ε.