Equidistribution in the space of 3-lattices and Dirichlet-improvable vectors on planar lines

Let X = SL3(R)/SL3(Z), and gt = diag(e2t,e−t,e−t). Let ν denote the push-forward of the normalized Lebesgue measure on a segment of a straight line in the expanding horosphere of {gt}t>0, under the map h 7→ hSL3(Z) from SL3(R) to X. We give explicit necessary and sufficient Diophantine conditions on the line for equidistribution of each of the following families of measures on X:

(1) gt-translates of ν as t → ∞.

(2) averages of gt-translates of ν over t ∈ [0, T ] as T → ∞.

(3) gti -translates of ν for some ti → ∞.

We apply this dynamical result to show that Lebesgue-almost every point on

the planar line y = ax + b is not Dirichlet-improvable if and only if (a, b) ∈/ Q2 .