Dense sets of integers with prescribed representation functions

Let A be a set of integers and let h \geq 2. For every integer n, let r_{A, h}(n) denote the number of representations of n in the form n=a_1+...+a_h, where a_1,...,a_h belong to the set A, and a_1\leq ... \leq a_h. The function r_{A,h} from the integers Z to the nonnegative integers N_0 U {\infty} is called the representation function of order h for the set A. We prove that every function f from Z to N_0 U {\infty} satisfying liminf_{|n|->\infty} f (n)\geq g is the representation function of order h for some sequence A of integers, and that A can be constructed so that it increases "almost" as slowly as any given B_h[g] sequence. In particular, for every epsilon >0 and g \geq g(h,epsilon), we can construct a sequence A satisfying r_{A,h}=f and A(x)\gg x^{(1/h)-epsilon}.Comment: 10 page

Some results on rational surfaces and Fano varieties

The goal of this article is to study the equations and syzygies of embeddings of rational surfaces and certain Fano varieties. Given a rational surface X and an ample and base-point-free line bundle L on X, we give an optimal numerical criterion for L to satisfy property Np. This criterion turns out to be a characterization of property Np if X is anticanonical. We also prove syzygy results for adjunction bundles and a Reider type theorem for higher syzygies. For certain Fano varieties we also prove results on very ampleness and higher syzygies.Comment: 26 pages, AMSTe

On the monotonicity of the correction term in Ramanujan's factorial approximation

We present two new proofs of the monotonicity of the correction term $\theta_n$ in Ramanujan's refinement of Stirling's formula.Comment: Latex, 5 page

Compatibility of CAST search with axion-like interpretation of PVLAS results

The PVLAS collaboration has results that may be interpreted in terms of a light axion-like particle, while the CAST collaboration has not found any signal of such particles. We propose a particle physics model with paraphotons and with a low energy scale in which this apparent inconsistency is circumvented.Comment: 10 pages, 3 figures, version accepted for publication in Physical Review Letter