Claims

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1958_01_01_erdos_szusz_turan: Theorem III of the 1958 paper that posed the problem proves that the limit of S(N,A,c) exists and equals 12A log c/pi^2 whenever 0<A<c/(1+c^2), the range in which the approximation intervals do not overlap.

1962_05_01_kesten: Theorem 2 of Kesten's 1962 paper proves that the limit of S(N,A,c) exists for Ac<=1 and gives closed forms on c/(1+c^2)<=A<=min(1/2,1/c) and on 1/2<=A<=1/c, extending the proposers' sparse-range value.

1966_01_01_kesten_sos: Kesten and Sós show in 1966 that the measure S(N,A,c) converges as N grows, for every A>0 and c>=1, without finding the value of the limit; this is the existence half of the question.

2006_01_01_xiong_zaharescu: Xiong and Zaharescu reprove in 2006 that the limiting measure exists for every A>0 and c>=1 and give a formula for it as a finite alternating sum of double integrals, from which the closed forms in the known ranges follow.

2008_08_01_boca: Boca's 2008 paper proves, according to its abstract, that the limiting measure S(N,A,c) exists and identifies it for all A>0 and c>1.

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