Existence and value of the survivor-set growth constant

Determine whether the limit of log N(n) divided by n/log n exists, and, if it exists, prove or disprove that its value is log 2; equivalently, resolve the corresponding exponential growth constant for the number N(n) of distinct survivor sets.

Background

The paper proves the bounds log 2 ≤ liminf log N(n)/(n/log n) ≤ limsup log N(n)/(n/log n) ≤ 2 log 2. Thus the logarithmic growth of N(n) is known only within a factor-two window for the leading constant.

The main comparison theorem shows that log N(n) is exponentially equivalent, up to an error o(n/log n), to log A_adm(n), where A_adm(n) counts prime-admissible subsets. Consequently, determining the limit for N(n) is equivalent at this scale to the unresolved block-complexity problem for the prime-admissible subshift. The paper also notes that the conjectural lower endpoint log 2 would imply the sharp asymptotic lower bound H(k)~k log k for the diameter of admissible k-tuples.