Explain the apparent regularity in the Ulam sequence

Explain the observed distribution of the sequence αa_n modulo 2π for the Ulam sequence, where α is the numerically suggested real number α ≈ 2.57145.

Background

The paper discusses numerical evidence suggesting that the Ulam sequence has more structure than its apparent irregularity indicates. In particular, one author observed a real number α ≈ 2.57145 such that the sequence αa_n modulo 2π exhibits a particularly curious distribution. Although this phenomenon has motivated subsequent research, the paper states that it remains unexplained.

References

Around ten years ago, one of the authors discovered (numerically) that the ‘great irregularity’ already observed by Queneau seems to exhibit a great degree of regularity: there seems to exist a real number α ∼ 2.57145 . . . so that the sequence αa_n mod 2π exhibits a most curious distribution [23]. This inspired subsequent work [1, 2, 3, 11, 12, 13, 14, 15, 22] but remains unexplained to this day.

— Small gaps in the Ulam sequence  (2501.16285 - Clément et al., 27 Jan 2025) in Section 1.1, page 1