[seqfan] Definition of A052130; question about Beatty sequences.

[L. Edson Jeffery]

Can someone please explain why the condition "For m very large" is required
in the definition (%N) of A052130 (https://oeis.org/A052130)?

For adjacent terms in a Beatty sequence, are there always only two distinct
first differences, and is there a formula for determining the maximum
distance between adjacent terms?

Ed Jeffery