Erdos 1958 remarks theory diophantine approximation

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P. Erdős, P. Szüsz, P. Turán: Remarks on the theory of diophantine approximation, Colloq. Math. 6 (1958), 119--126 (MR 21 #1290; Zentralblatt 87, 43).

The authors study the localization problem for |a - x/y| <= A/y^2 with 0 < a < 1 and coprime integers x, y with y > 1, asking for the measure of the set S(N,A,c) of those a for which the inequality is solvable with N <= y <= cN. Contrary to the guess that the constraint forces measure zero, Theorem I gives lim inf |S(N,A,c)| >= (3/pi^2)(1 - 1/c^2) min(1, 2A) for A > 0 and c > 1, and Theorem II sharpens this to (3/pi^2)(5/4 - 2/c^2) for A >= 1 and c >= 2. Theorem III establishes that for 0 < A < c/(1+c^2) the limit exists and equals f(A,c) = 12 A log c / pi^2, while Theorem IV shows |S(N,A,c)| < 1 - 1/(40 A^4 c^4 pi) for A > 10 and c > 10 and all large N (p. 120), so the limit, if it exists, is below 1. Whether f(A,c) exists for all A > 0 and c > 1 is posed as Problem I (P 241), and Problem II (P 242) asks the analogous question for R(N,c), the set of a whose regular continued fraction has a denominator q_v in [N, cN], with Theorem I at A = 1/2 giving the first step. Problem 1001 is Problem I (P 241), the existence and form of f(A,c); the paper supplies the theorems above as partial results on it and states both open problems P 241 and P 242.

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Bears on. #1001: the problem is the paper's Problem I (P 241); Theorems I and II give positive lower bounds for lim inf |S(N,A,c)|, Theorem III gives the limit for 0 < A < c/(1+c^2), and Theorem IV bounds |S(N,A,c)| below 1 for A > 10, c > 10; the paper leaves the existence of f(A,c) open outside the range of Theorem III.

Results to transcribe.

• Theorem I: For A > 0 and c > 1, lim inf_{N to infinity} |S(N,A,c)| >= (3/pi^2)(1 - 1/c^2) min(1, 2A) (p. 120; Theorems I and II are printed with lim, read as lim inf since the limit is not known to exist).
• Theorem II: For A >= 1 and c >= 2, lim inf_{N to infinity} |S(N,A,c)| >= (3/pi^2)(5/4 - 2/c^2).
• Theorem III: For 0 < A < c/(1+c^2) the limit f(A,c) = lim |S(N,A,c)| exists and equals 12 A log c / pi^2.
• Theorem IV: For A > 10 and c > 10 and all sufficiently large N, |S(N,A,c)| < 1 - 1/(40 A^4 c^4 pi), so f(A,c) < 1 if it exists.
• Problem I (P 241): Does lim_{N to infinity} |S(N,A,c)| exist for all A > 0 and c > 1, and if so what is its explicit form?
• Problem II (P 242): For R(N,c) the set of a in (0,1) whose regular continued fraction has a denominator q_v in [N, cN], does lim |R(N,c)| = Phi(c) exist, and what is it?