Eric Schmutz

Rational points on the unit sphere

AbstractIt is known that the unit sphere, centered at the origin in ℝn, has a dense set of points with rational coordinates. We give an elementary proof of this fact that includes explicit bounds on the complexity of the coordinates: for every point ν on the unit sphere in ℝn, and every ν > 0; there is a point r = (r1; r2;…;rn) such that: ⊎ ‖r-v‖∞ < ε.⊎ r is also a point on the unit sphere; Σ ri2 = 1.⊎ r has rational coordinates;

A central limit theorem on gln(fq)

r_i = \frac{{a_i }} {{b_i }}

The maximum degree in a random tree and related problems

for some integers ai, bi.⊎ for all

Gap-Free Set Partitions

i,0 \leqslant \left| {a_i } \right| \leqslant b_i \leqslant (\frac{{32^{1/2} \left\lceil {log_2 n} \right\rceil }} {\varepsilon })^{2\left\lceil {log_2 n} \right\rceil }

The order of a typical matrix with entries in a finite field

. One consequence of this result is a relatively simple and quantitative proof of the fact that the rational orthogonal group O(n;ℚ) is dense in O(n;ℝ) with the topology induced by Frobenius’ matrix norm. Unitary matrices in U(n;ℂ) can likewise be approximated by matrices in U(n;ℚ(i))

Random Set Partitions

Abstract We prove a central limit theorem for the number of different part sizes in a random integer partition. If λ is one of the P(n) partitions of the integer n, let Dn(λ) be the number of distinct part sizes that λ has. (Each part size counts once, even though there may be many parts of a given size.) For any fixed x, #(λ: D n (λ) ⩽ A n + xB n } P(n) → 1 2π ∫ −∞ x l −t 2 2 dt as n → ∞, where A n = (√6/π)n 1 2 and B n = (ρ6/2π − √54/π 3 ) 1 2 n 1 4 .