(Order of) multiplicity of primefactor 2 in the integers near $N$'th root of $2^S$?

Question

Consider the $N$'th root of $2^S$ . This roots are noninteger (irrational) when $S/N$ is noninteger. To avoid duplicates consider in the following only $S$ for a given $N$ when $\gcd(N,S)=1$.
Let's assume $N, S\in \mathbb N^+$, $S>N$ (and $\gcd(S,N)=1$ when we want avoid duplicates). Let's denote $R=S/N$ and assume $S,N$ such that this rational value $R>1$. Let's then denote $\rho = 2^R$ .

Now I am interested in the multiplicity of the primefactor $2$ in the neighboured integers, say $m_l = \lfloor \rho \rfloor $ and $m_u = \lceil \rho \rceil = m_l+1 $. Because only one can have the primefactor $2$ we can introduce $$ A_{N,S} = \nu_2 (m_l \cdot m_u) $$
where $\nu_2(\cdot)$ means the valuation for the primefactor $2$.

Using small $N$ and looking at $S$ up to $1000$, heuristics seem to indicate that $A$ is roughly of logarithmic order depending on $S$.

Can we have an upper bound for $A$ depending on $S$ (if we keep $N$ constant)?

The heuristics remind me a bit of the primefactors $p^k$ in the canonical primefacorization of the cyclotomic expressions $a^n-1 $ depending on increasing $n$ - however it seems much more random.

update I've found an article of Bailey, Borwein, Crandall, Pomerance "On the binary representation of algebraic numbers" (2003?) which deals with this and more general problems. I found them mention that "for large N cannot have a zero-run of more than $2\ln N$ consecutive zeros" which is very near to what I am considering here. (chap 11, about "open problems", pg 29 of my pdf-preprint)

Answer 1

This answer is not yet complete, but it gives already a very instructive "formula" for determination of $A$ .
The main ingredient for the formula is the binary representation of $2^{1/N}$

• Let us assume $N=2$. The binary representation of $\rho=2^{1/2}$ is "1.01101010000010011110011001100111..."_2
  The valuations $\nu_2(\lfloor 2^{k-1} \rho \rfloor)$ represented as string is s_l="10010101234501200001200120012000..." . Accordingly we can determine s_u="01201010000010012340012001200123..." for the valuations of the ceilings. Having such a compact representation we can get
  __"10010101234501200001200120012000..." s_l (valuations in floors) counting consecutive zeros
  "1.01101010000010011110011001100111..."_2 bitstring for $2^{1/2}$
  __"01201010000010012340012001200123..." s_u (valuations in ceils) counting consecutive ones
  We can immediately see, how the exponents at primefactor $2$ in $w_l(k) = \lfloor 2^{k-1} \cdot 2^{1/2} \rfloor$ occur if we follow the zeros in the bitstring and look in the according "digit" in $s_l$ above it.
  Completely analoguous is it with the ceilings.

Of course this is not yet a full answer, how large $A(S,N)$ can become depending on $S$ and $N$. However it indicates easily how it depends on the approximation of $2^k \cdot 2^{1/N}$ to the unit, and this obviously encoded in the bitstring-representation of $\rho_N=2^{1/N}$ Thus the Baker's approximation-rules occur here. As we deal with $N'th$ roots of $2$ here, we'll find not very well diophantine approximations as that fractions are irrational, but still algebraic, numbers. I think I've seen research on the numbers of consecutive zeros (or ones) in the bitstring of irrational numbers (however maybe of $\log_2(3)$ instead, have it not at hand) but I think it is reasonable guess, that by this $A(S,N)$ grows at most logarithmically with perhaps finitely many exceptions.

One more example:

• Let us assume $N=3$. The binary representation of $\rho=2^{1/3}$ is "1.01000010100010100010111110011000..."_2
  The valuations $\nu_2(\lfloor 2^{k-1} \rho \rfloor)$ represented as string is s_l="10123401012301012301000001200123..." . Accordingly we can determine s_u="01000010100010100010123450012000..." for the valuations of the ceilings. Having such a compact representation we can write
  __"10123401012301012301000001200123..." s_l (valuations in floors) counting consecutive zeros
  "1.01000010100010100010111110011000..."_2 bitstring for $2^{1/3}$
  __"01000010100010100010123450012000..." s_u (valuations in ceilings) counting consecutive ones