2014/10/10 by O'Bryant, Kevin
#11B83 #11J70 #11J99 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1410.2927
We study the function M(t,n) = Floor[ 1 / t^(1/n) ], where t is a positive real number, Floor[.] and . are the floor and fractional part functions, respectively. In a recent article in the Monthly, Nathanson proved that if log(t) is rational, then for all but finitely many positive integers n one has M(t,n) = Floor[ n / log(t) - 1/2 ]. We extend this by showing that, without condition on t, all but a zero-density set of integers n satisfy M(t,n) = Floor[ n / log(t) - 1/2 ]. Using a metric result of Schmidt, we show that almost all t have asymptotically log(t) log(x)/12 exceptional n