The density of rational points on curves and surfaces

Let $C$ be an irreducible projective curve of degree $d$ in $\mathbb{P}^3$, defined over $\overline{\mathbb{Q}}$. It is shown that $C$ has $O_{\varepsilon,d}(B^{2/d+\varepsilon})$ rational points of height at most $B$, for any $\varepsilon>0$, uniformly for all curves $C$. This result extends an estimate of Bombieri and Pila [Duke Math. J., 59 (1989), 337-357] to projective curves. For a projective surface $S$ in $\mathbb{P}^3$ of degree $d\ge 3$ it is shown that there are $O_{\varepsilon,d}(B^{2+\varepsilon})$ rational points of height at most $B$, of which at most $O_{\varepsilon,d}(B^{52/27+\varepsilon})$ do not lie on a rational line in $S$. For non-singular surfaces one may reduce the exponent to $4/3+16/9d$ (for $d=4$ or 5) or $\max\{1,3/\sqrt{d}+2/(d-1)\}$ (for $d\ge 6$). Even for the surface $x_1^d+x_2^d=x_3^d+x_4^d$ this last result improves on the previous best known. As a further application it is shown that almost all integers represented by an irreducible binary form $F(x,y)\in\mathbb{Z}[x,y]$ have essentially only one such representation. This extends a result of Hooley [J. Reine Angew. Math., 226 (1967), 30-87] which concerned cubic forms only. The results are not restricted to projective surfaces, and as an application of other results in the paper it is shown that $\#\{(x_1,x_2,x_3)\in\mathbb{N}^3:x_1^d+x_2^d+x_3^d=N\} \ll_{\varepsilon,d} N^{\theta/d+\varepsilon}$ with $\theta=\frac{2}{\sqrt{d}}+\frac{2}{d-1}.$ When $d\ge 8$ this provides the first non-trivial bound for the number of representations as a sum of three $d$-th powers

Finite Sample Bernstein -- von Mises Theorem for Semiparametric Problems

The classical parametric and semiparametric Bernstein -- von Mises (BvM) results are reconsidered in a non-classical setup allowing finite samples and model misspecification. In the case of a finite dimensional nuisance parameter we obtain an upper bound on the error of Gaussian approximation of the posterior distribution for the target parameter which is explicit in the dimension of the nuisance and target parameters. This helps to identify the so called \emph{critical dimension} $p$ of the full parameter for which the BvM result is applicable. In the important i.i.d. case, we show that the condition "$p^{3} / n$ is small" is sufficient for BvM result to be valid under general assumptions on the model. We also provide an example of a model with the phase transition effect: the statement of the BvM theorem fails when the dimension $p$ approaches $n^{1/3}$. The results are extended to the case of infinite dimensional parameters with the nuisance parameter from a Sobolev class. In particular we show near normality of the posterior if the smoothness parameter $s$ exceeds 3/2

Mean values with cubic characters

We investigate various mean value problems involving order three primitive Dirichlet characters. In particular, we obtain an asymptotic formula for the first moment of central values of the Dirichlet L-functions associated to this family, with a power saving in the error term. We also obtain a large-sieve type result for order three (and six) Dirichlet characters.Comment: 22 pages; greatly shortened, simplified and corrected versio