In this talk, we explain what is the sum–product phenomenon for sets of positive density in the integer lattice and show how it can be obtained via cancellation estimates for certain exponential sums. The polynomial analogue of Bogoljuboff’s theorem provides one example of such a phenomenon — for any integer valued polynomial p of degree at least two with $p(0)=0$, and for any $\delta>0$, there exists $k(\delta)$ such that, for every set $E$ in $\mathbb{Z}^2$ with upper density at least $\delta$, the set $$ { x + p(y) : x,y \in E-E } $$ contains a subgroup $k\mathbb{Z}$ for some $k <= k(\delta)$. We will present the proof of this theorem which uses Hua bounds and Weyl’s equidistribution. The talk is based on joint works with M. Björklund and K. Bulinski.
Measuring the set of simultaneously well approximable points on manifolds is one of the most intricate problems in metric theory of Diophantine approximation. Unlike the dual case of well approximable linear forms, the results here are known to depend on a manifold. For example, some of the manifolds do not contain simultaneously very well approximable points at all, while for the others the set of such points always has positive Hausdorff dimension. In this talk, we will closely look at the Veronese curve $\{x, x^2, x^3, \ldots, x^n\}$, discuss what is known about the sets of simultaneously well approximable points on it and provide several new results. In particular, we provide the Hausdorff dimension of the set of $x$ such that $\lambda_n(x)$ lies either close enough or far enough from $1/n$.
A question going back to Halmos asks when two approximately commuting matrices of a certain kind are close to genuinely commuting matrices of the same kind. It was quickly realized that dimension-independent results were far more difficult to obtain, and in work of Lin the result was settled for self-adjoint matrices. Estimates for the distance to commuting self-adjoint matrices were sought out by many authors until optimal bounds were established by Kachkovskiy and Safarov. On the other hand, it has long been known, since the work of Voiculescu, that there is an obstruction for dimension-independent approximately commuting unitary matrices to be close to commuting unitary matrices. However, in work of Gong and Lin, and of Eilers, Loring and Pedersen, it was shown that when this obstruction vanishes a positive result still holds. Despite this, quantitative bounds for the distance in terms of the commutator of the unitary matrices were still unknown. In this talk I will report on joint work with Hall and Kachkovskiy where we show that under the vanishing of said obstruction, we can find bounds for the distance to commuting unitary matrices in terms of the commutator of the original pair of unitary matrices.
Colour two circles in the same way by making an arc red and the rest green. Start somewhere on one of the circles. At each step, rotate by a fixed angle $\alpha$. If the image lands on a red point, stay put at that point. If it lands on a green point, move to the corresponding point on the other circle. This gives rise to a sequence of points on the two circles. What can we say about the distribution of this sequence?
How does the height of an elliptic curve change when passing through an isogeny? What if we average over all images under a Hecke correspondence? What even is the height of an elliptic curve? In this talk I answer these questions.
Palindromes; numbers which are the same when written forwards and backwards, have been studied since antiquity. Prior to the 21st Century, palindromes were only studied as part of recreational and elementary number theory. However, in recent years, studying their multiplicative structure using deep analytic methods has been a hot topic. In this talk, I will discuss recent work with Bryce Kerr which proves existence of infinitely many square-free palindromes, along with a corresponding asymptotic formula. This resolves an open problem of Igor and his collaborators. This will be a two-part talk, with further details given in the following talk by Bryce.
Palindromes; numbers which are the same when written forwards and backwards, have been studied since antiquity. Prior to the 21st Century, palindromes were only studied as part of recreational and elementary number theory. However, in recent years, studying their multiplicative structure using deep analytic methods has been a hot topic. In this talk, I will discuss recent work with Daniel Johnston which proves existence of infinitely many square-free palindromes, along with a corresponding asymptotic formula. This resolves an open problem of Igor and his collaborators. This will be part II of Daniel's talk.
In this talk I will discuss recent results on the Skolem Problem for specialisations of linear recurrences defined over a function field. In particular, I will report on work in progress with Philipp Habegger and David Masser, which gives a finiteness result for specialisations at roots of unity. I will also make connection to certain G.C.D. problems for linear recurrences.
Following the celebrated work of Herr and Kwak, I will explain a multilinear argument in the proof of their $L^4$ exponential sum estimate. It suggests interesting questions about counting solutions to Diophantine systems where each variable takes value in a different set. I will reflect upon those and discuss the role of off-diagonal solutions.
In this talk, we discuss several counting questions on the denominators of rational points on elliptic curves that have some special behaviour, such as being perfect powers. Our main focus is on estimating the frequency of tuples of rational points, whose denominators or their $x$-coordinates satisfy multiplicative relations, both for the points of the form $nP+Q$, and for the points of bounded canonical height. This is based on a joint work with Attila Berczes, Lajos Hajdu, Alina Ostafe, and Igor E. Shparlinski.
In joint work with Alexander Fish (University of Sydney), we introduce a notion of Kloosterman-type sums associated to an arbitrary countably infinite field and investigate their asymptotic behaviour from an ergodic-theoretic perspective. In this talk I will survey the main ideas and results of this work, and explain how these sums can be used to derive sum–product phenomena in infinite fields. Along the way, I will highlight several features that sharply distinguish these objects from their classical finite-field counterparts, revealing new and somewhat unexpected behaviour.
In this talk, I will compare the asymptotic and uniform Diophantine approximation and show how both can be naturally and effectively described using continued fraction expansions. We obtain a near-complete characterisation of the metric properties of the associated sets.
Two matrices are called equivalent if one can be transformed into the other by multiplying with invertible matrices on the left and right. Extending this idea to 3-tensors, it is natural to define two 3-tensors as isomorphic if they can be transformed into each other by multiplication with three invertible matrices along the three directions. The problem of testing isomorphism of tensors over finite fields naturally connects to some of Igor’s wide-ranging research interests: algorithms, pseudorandomness, cryptography, and interactions between combinatorics and algebra. In this talk I will report on recent works and our current understanding of tensor isomorphism over finite fields.
Shparlinski and his collaborators have in several papers studied distributional questions about random matrices as well as about class numbers and $L$-values. Connecting these two themes, we report on work with Prajeet Bajpai and work in progress with Ross Paterson and Sameera Vemulapalli on investigating some archimedean aspects of class groups and unit groups of number fields using random matrices, in the spirit of the Cohen-Lenstra heuristics.
In this talk, I will present a central limit theorem describing the fluctuations of the number of zeros (local statistics) of various families of $L$-functions around their mean. The correlations of these fluctuations coincide with those obtained by Wieand, Diaconis, and Evans for the number of eigenvalues of random matrices. The families of $L$-functions considered here are defined over function fields associated with hyperelliptic curves of large genus over a fixed finite field.
In this talk on the occasion celebrating the mathematical author of myriad worthy and enduring manuscripts, I shall report on some recent work, jointly with P. Gao, evaluating the first moment of the family of primitive quadratic Hecke $L$-functions in the Gaussian field using the method of double Dirichlet series under the Riemann hypothesis and the Lindelöf hypothesis. We obtain asymptotic formulas with secondary main terms and error terms of size that is one quarter of that of the main term.