Effective equidistribution of expanding horospheres in the locally symmetric space SO( d ) ∖ SL d (ℝ) / SL d (ℤ)

We use an equivalence between counting lattice points in dilated d -dimensional ellipsoids (Euclidean counting) and counting lifts of a closed horosphere intersecting balls of increasing radii to obtain two types of results. First, using an <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> {L²} -integral error estimate for Euclidean counting, we prove effective equidistribution results for a family of expanding horospheres in the locally symmetric space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mi>SO</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>d</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>\</m:mo> <m:msub> <m:mi>SL</m:mi> <m:mi>d</m:mi> </m:msub> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ℝ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>/</m:mo> <m:msub> <m:mi>SL</m:mi> <m:mi>d</m:mi> </m:msub> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ℤ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {SO(d)d({R})/SLd({% Z})} . Secondly, we derive from uniform error estimates in Euclidean counting, error estimates for counting <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>SL</m:mi> <m:mi>d</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ℤ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {{{ SL}d({Z})}} -orbit points in a certain increasing family of subsets in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mi>SO</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>d</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>\</m:mo> <m:msub> <m:mi>SL</m:mi> <m:mi>d</m:mi> </m:msub> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ℝ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {SO(d)d({R})} (which we call truncated chimneys ), and for counting the number of lifts of a closed horosphere that intersect a ball with large radius.