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Density and equidistribution of integer points

Gorodnyk, Oleksandr

Abstract Details

2003, Doctor of Philosophy, Ohio State University, Mathematics.

Celebrated work of G. Margulis on the conjecture of Oppenheim has established that for a real nondegenerate indefinite quadratic form Q in dimension d≥3 , which is not a scalar multiple of a rational form, the set Q(Zd) is dense in R . We extend this result to the case of a pair consisting of a quadratic form Q and a linear form L . Namely, we show that the set {(Q(x),L(x)):x∈Zd}⊂R2 is dense in R2 provided that some natural algebraic conditions are satisfied.

We also consider a similar, but much more complicated problem concerning values of a system of quadratic forms at integer points. Let Qi, i=1,…,s, be nondegenerate indefinite quadratic forms of dimension d. Under some conditions on the intersection of zero surfaces {Qi=0}, i=1,…,s, we compute measure and Hausdorff dimension of the set of g∈ GL(d,R) such that the set {(Q1(gx),…,Qs(gx)):x∈Zd}⊂Rs accumulates at (0,…,0). In some cases, this Hausdorff dimension is fractional.

Using recent fundamental results on rigidity of unipotent flows, we investigate distribution of orbits of a lattice (of a Lie group) in a homogeneous space. More precisely, let G be a matrix Lie group, Γ a lattice in G, and H a noncompact closed subgroup of G. We consider the action of Γ on G/H. Fix a norm ||·|| on G. For T>0, Ω⊆G/H, and x0∈G/H, define N T (Ω,x 0)=#{γ∈Γ:γ·x 0 ∈Ω,||γ||<T}. Distribution of Γ-orbits in G/H is reflected in the asymptotic behavior of the quantity NT (Ω,x0) as T→∞. We determine this asymptotic behavior in several cases:

A lattice in SL (d,R) acting on the Furstenberg boundary of SL (d,R);

A lattice in SL (d,R) acting on the space of k-frames in Rd (1≤k≤d-1);

A lattice in a general matrix Lie group G acting on a factor space G/H where H is either semisimple or unipotent subgroup of G that satisfies some technical conditions.

Also we deduce some interesting corollaries concerning distribution of values of the Gram matrix of a quadratic form on integer frames and uniform distribution of orbits of SL(d,Z) in the torus Td.

Vitaly Bergelson (Advisor)
231 p.

Recommended Citations

Citations

  • Gorodnyk, O. (2003). Density and equidistribution of integer points [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1054487714

    APA Style (7th edition)

  • Gorodnyk, Oleksandr. Density and equidistribution of integer points. 2003. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1054487714.

    MLA Style (8th edition)

  • Gorodnyk, Oleksandr. "Density and equidistribution of integer points." Doctoral dissertation, Ohio State University, 2003. http://rave.ohiolink.edu/etdc/view?acc_num=osu1054487714

    Chicago Manual of Style (17th edition)