Let $X\subset {ℝ}^n$ be a compact subanalytic surface. This paper shows that, in a suitable sense, there are very few rational points of $X$ that do not lie on some connected semialgebraic curve contained in $X$.

It 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 = (r 1; r 2;…;r n) such that: ⊎ ‖r-v‖∞ < ε.⊎ r is also a point on the unit sphere; Σ r i 2 = 1.⊎ r has rational coordinates; $$r_i=\frac{a_i}{b_i}$$ for some integers a i, b i.⊎ for all $$i,0⩽\left|a_i\right|⩽b_i⩽{\left(\frac{{32}^{1/2}⌈lo{g}_{2}n⌉}{\epsilon }\right)}^{2⌈lo{g}_{2}n⌉}$$ . One consequence of this...