Xiong 2006 problem erdos szusz turan diophantine

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Xiong, Maosheng and Zaharescu, Alexandru, A problem of {E}rdős-Szüsz-Turán on {D}iophantine approximation. Acta Arith. 125 (2006), no. 2, 163--177.

The paper concerns S(m,alpha,c), the set of xi in [0,1] admitting integers a,q with m <= q <= mc, gcd(a,q) = 1 and |q xi - a| <= alpha/q, whose measure Erdos, Szusz and Turan computed as (12 alpha/pi^2) log c when alpha <= c/(1+c^2), leaving open whether the limit exists in general. Theorem 2 (Section 4) gives a new proof that rho(alpha,c) = lim_m mu(S(m,alpha,c)) exists for all alpha > 0 and c >= 1, together with explicit formulas for computing it; this recovers and extends the formulas of Kesten (valid for alpha c <= 1) and the existence result of Kesten and Sos. Theorem 1 is the new distributional statement: for any subinterval I of [0,1], the restricted sets S_I(m,alpha,c) satisfy lim_m mu(S_I(m,alpha,c)) = |I| rho(alpha,c), so the limiting mass is uniformly distributed across [0,1]. The method recasts the problem in terms of how visible lattice points subject to congruence conditions are spaced, which leads to counting modular inverses in residue classes and to Kloosterman sum estimates. This is the reference for problem 1001, the Erdos-Szusz-Turan diophantine approximation problem.

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Bears on. #1001

Results to transcribe.

• theorem_1: For any alpha > 0, c >= 1 and any subinterval I of [0,1], lim_{m->inf} mu(S_I(m,alpha,c)) exists and equals |I| rho(alpha,c).
• theorem_2: New proof that rho(alpha,c) = lim_{m->inf} mu(S(m,alpha,c)) exists for all alpha > 0 and c >= 1, with explicit formulas for its computation.