Claims

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1998_04_01_erdos_komornik: The 1998 theorem that the gaps of the ordered sums of distinct powers of q tend to zero for every q in (1, 2^(1/4)] other than the square root of the second Pisot number, the first resolution; refereed, credited by the site.

2011_03_23_akiyama_komornik: The theorem that the gaps of the ordered sums of distinct powers of q tend to zero for every q in (1, 2^(1/3)], closing the point Erdős and Komornik had left out; refereed in the Journal of Number Theory (2013).

2011_11_10_feng: Feng's theorem that the gaps tend to zero for every q in (1, sqrt 2) whose square is not a Pisot number; with Siegel's smallest Pisot number this answers the question for every q below about 1.151; refereed in JEMS 2016.

2026_04_16_acosta_de_leon: A two-page note of 16 April 2026, posted on a data repository and linked from the problem's thread, that deduces the answer for every q below the square root of the smallest Pisot number from Feng's Theorem 1.4; pending.

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