Edit on July 31: Now the upper bound is not a complete answer tight (up to replacing n by O(n)). The improvement over the question, but here older version is in the argument after Lemma 1. Now we consider the Weil absolute logarithmic height of an algebraic number instead of the length.
Here is a proof that D(n) is at most doubly-exponential with quadratic exponent, i.e., D(n) < 22cn2 for some positive constant c>0 c>0 for sufficiently large n. In other words, the answer to the “can one do better” part of the question 2 is negative.
As Terry Tao pointed out, this problem can be rephrased in the algebraic form. We have z0=0 and z1=1, and any zn (n≥2) can be obtained from earlier numbers by addition, subtraction, multiplication, division or square root. The claim follows if |zn| < 22O(n2).
The length of an algebraic number α
There is the sum of a function called the Weil absolute logarithmic height h(α) defined on algebraic numbers α which takes nonnegative real values. See Section 3.2 of [Wal00] for its definition and the coefficients proof of the integer-coefficient minimum polynomial of α.
Lemma 2following properties:
Proof idea: Construct the minimum polynomials of α+β etc. by using resultants,
By combining Lemmas 1 and 2 and using the properties 2–5 and the mathematical induction, we obtain can prove that the length of zh(zn is at most 2) ≤ 2n2+n+1 = 22O(n ln 2).
Since the length is an upper bound on By combining the height property 1 and the absolute value of a root of a polynomial is at most Lemma 1plus the height of the polynomial, we obtain that |zn| is also at most 2≤ 2O(n2)2n.
[Wal00] Michel Waldschmidt: I do not claim that the bound in Lemma 2 is close to optimal. However, even if Diophantine Approximation on Linear Algebraic Groups: Transcendence Properties of the bound Exponential Function in Lemma 2 is improved to, say, L√(d), the upper bound on D(n) we obtain is still 22O(n2). In particularSeveral Variables, if D(n) < 22cn for some constant c>0 for sufficiently large nSpringer, its proof cannot be obtained by improving the bound in Lemma 2 alone2000.