list = {};
num = 10000;
d = 1;
alpha = 1.0 Pi;
For[n = 1, n ≤ num, n++, list = AppendTo[list, Mod[n^d alpha, 1]]];
list = Sort[list];
Histogram[list, Automatic, "Probability"]
diff = {};
For[n = 1, n ≤ num-1, n++, diff = AppendTo[diff, (1.0 list[[n+1]] - 1.0 list[[n]]) * num]];
Histogram[diff, Automatic, "Probability"]

list

diff