An integer-visiting process with real multipliers: coverage and rates

A positive noninteger start has its integer part and zero visited; its fractional part is repeatedly multiplied by a real β > 1, visiting and subtracting each unvisited integer part. For every β > 1 and almost every start (or every start and almost every β), this process visits every positive integer, and all limit points of cov (t)/(t log t) lie in [2(β − 1)/(2β + 1), 2β(β − 1)/(2β − 1)], where cov (t) counts new visits until 1, …, t are visited; r new visits take r log_β r + O(r) multiplications, and their largest label is within constant factors of rβ^(√(2 log_β r))/√(log_β r). At integer multipliers its base-β logarithm has the centring of symmetric digital-search-tree height, proved directly in a companion paper. The lower and upper limits of cov (t)/(t log t) are determined exactly for integer multipliers. With 0, 1 visited, a computer-assisted proof gives a residual in ℤ[β_gold] never visiting 99 under β_gold = (1 + √(5))/2; coverage from π at β = 2 and with start and multiplier β_gold remains open. MSC 2020: Primary 11K55; Secondary 11K16, 37E05, 11A63, 60F15, 68P05, 68P30. The accompanying files contain the manuscript, its LaTeX source, and the programs, exact certificates and numerical data for the trajectories, geometric partitions, completion times and finite-sample comparisons.