Related papers: Bourgain-type projection theorems over finite fiel…

We prove an analogue of the prime number theorem for finite fields.

We prove an inequality that extends to arbitrary measures the hyperplane inequality for volume of unconditional convex bodies originally observed by Bourgain.

We propose several techniques to construct complete permutation polynomials of finite fields by virtue of complete permutations of subfields. In some special cases, any complete permutation polynomials over a finite field can be used to…

This is an expository note discussing how the Erdos--Ramanujan proof of Bertrand's postulate may be adapted to show the existence of finite fields.

We prove the Chern-Gauss-Bonnet Theorem using sigma models whose source supermanifolds have super dimension 0|2. Along the way we develop machinery for understanding manifold invariants encoded by families of 0|n-dimensional Euclidean field…

Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we develop general theorems on permutation polynomials over finite fields. As a…

We provide a characterisation of differentially large fields in arbitrary characteristic and a single derivation in the spirit of Blum axioms for differentially closed fields. In the case of characteristic zero, we use these axioms to…

In this paper, we introduce a new homological invariant called quasi-projective dimension, which is a generalization of projective dimension. We discuss various properties of quasi-projective dimension. Among other things, we prove the…

We use spectral theory and algebraic geometry to establish a higher-degree analogue of a Szemer\'edi--Trotter-type theorem over finite fields, with an application to polynomial expansion.

The paper contains a construction of an analogue of the Fontaine-Wintenberger field-of-norms functor for higher dimensional local fields. This construction is done completely in terms of the ramification theory of such fields. It is applied…

The Gross-Kohnen-Zagier theorem describes Heegner points on a modular curve in terms of coefficients of modular forms. We give another proof of this theorem which generalizes to higher dimensions.

We show under general assumptions that the mean-field approximation for quan- tum many-boson systems is correct. Our contribution unifies and improves on most of the known results. The proof uses general properties of quantization in…

We prove various results on the size and structure of subsets of vector spaces over finite fields which, in some sense, have too many mutually orthogonal pairs of vectors. In particular, we obtain sharp finite field variants of a theorem of…

We show that Blokhuis' quadratic upper bound for two-distance sets is sharp over finite fields in almost all dimensions. Our construction complements Lison\v{e}k's higher-dimensional maximal constructions that were carried out in Lorentz…

We present a general analysis of gauge invariant, exact and metric independent forms which can be constructed using higher rank field-strength tensors. The integrals of these forms over the corresponding space-time coordinates provides new…

We estimate the proportion of function fields satisfying certain conditions which imply a function-field analogue of the Fontaine-Mazur conjecture. As a byproduct, we compute the fraction of abelian varieties (or even Jacobians) over a…

Marstrand's celebrated projection theorem gives the Hausdorff dimension of the orthogonal projection of a Borel set in Euclidean space for almost all orthogonal projections. It is straightforward to see that sets for which the Fourier and…

We study a Szemer\'edi-Trotter type theorem in finite fields. We then use this theorem to obtain an improved sum-product estimate in finite fields.

We study the problem of pointwise convegence for the Schr\"odinger operator on $\mathbb R^n$ along time sequences. We show that the sharp counterexample to the sequential Schr\"odinger maximal estimate given recently by Li, Wang and Yan…

We prove essentially optimal $L^p(\mathbb{R})$-estimates for variational variants of the maximal Fourier multiplier operators considered by Bourgain in his work on pointwise convergence of polynomial ergodic averages. As a corollary of our…