Discrepancy of the words ⌊(2n+k)r⌋ − ⌊(n+k)r⌋ − ⌊nr⌋ and twenty-two conjectures of Kimberling

For an irrational r > 0 and an integer k ≥ 1, the word s_k(n) = ⌊(2n+k)r⌋ − ⌊(n+k)r⌋ − ⌊nr⌋ takes only the values 0 and 1, each with density 1/2. In 2019 Kimberling added to the OEIS eleven such words, for r ∈ {φ, √2, √3, e} and k ≤ 4, together with the 22 sequences a(n) listing the positions of their 0s and 1s. For 21 of these he conjectured that 2n − a(n) is unbounded below and above, and for the remaining one that a(n) − 2n is unbounded above. We settle all 22 conjectures: 19 are true and 3 are false. The false ones all come from r = √2 with k ∈ {2, 4}: the partial sums of 2s_k(n) − 1 never exceed 0 (for k = 2) or 1 (for k = 4). Hence a(n) ≥ 2n for A327207, 2n − a(n) ≥ 1 for A327206, and 2n − a(n) ≥ 0 for A327223. For even k the qualitative statements follow, through a short reduction, from a theorem of Ying and Zheng on the one-sided boundedness of the local discrepancy D_N(α, 1/2), and also from earlier formulas of Roçadas and Schoißengeier, which Amoroso and Omarjee applied to r = e; for r ∈ {φ, √2, √3} they also follow from results of Boshernitzan and Ralston and of Dolgopyat and Sarig. The cases with odd k, where that reduction leads to a sum of two local discrepancies with irrational cut points, are new, and so are the exact bounds. For quadratic r we prove the results with Ostrowski-automatic sequences, verified in Walnut, and an exact block-sum analysis that also gives the growth rates of the extremes; for even k these rates also follow from known results on D_N(α, 1/2). For r = e we prove an exact formula for the discrepancy at N = t·q_i, where q_i are the convergent denominators of e, and obtain closed forms along N = t·q_{3j−2}.Version 2: the scripts blockformula.py and e_verify.py now exit with a nonzero status on any unexpected mismatch (e_verify.py still accepts the two cases with j = 1, which Theorem 10 excludes; see Remark 11), and the README now explains why the quadruple-precision floors for e in exact.c are exact (error bound in the Verification section of the paper). The paper is unchanged.