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Olof Sisask
Who?
I am an EPSRC postdoctoral research fellow at the School of Mathematical Sciences at Queen Mary, University of London. I completed my PhD under the supervision of Ben Green in 2009. I was a PhD student at the University of Bristol, though I spent most of my time as a long-term visitor at the University of Cambridge; I also spent the 2005–2006 academic year as a visiting graduate student at MIT.
I spent December 2010 – March 2011 visiting the University of British Columbia, and April – June 2011 at the Isaac Newton Institute for the Discrete Analysis programme.
I work mainly in an area of mathematics called additive combinatorics, a subject touching on combinatorics, analysis, probability and number theory. I am particularly interested in using ideas from analysis and probability to tackle problems of a combinatorial or number theoretical nature.
I spent December 2010 – March 2011 visiting the University of British Columbia, and April – June 2011 at the Isaac Newton Institute for the Discrete Analysis programme.
I work mainly in an area of mathematics called additive combinatorics, a subject touching on combinatorics, analysis, probability and number theory. I am particularly interested in using ideas from analysis and probability to tackle problems of a combinatorial or number theoretical nature.
Contact details
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Current research
I am interested in many questions in additive combinatorics, especially ones that relate to solutions to linear equations in subsets of Z/pZ. My main research topics are: Roth's theorem and its extensions; applications of analytic ideas to combinatorial problems; quadratic Fourier analysis; extremal structures in additive combinatorics and the structure of structure-avoiding sets; computational aspects of additive number theory. I am particularly interested in ways in which one can isolate the important aspects of some data and represent these using a small number of parameters.
Papers and articles- Arithmetic progressions in sumsets and Lp-almost-periodicity.With Ernie Croot and Izabella Laba.
- A removal lemma for linear configurations in subsets of the circle.With Pablo Candela. Accepted to Proc. Edinburgh Math. Soc.
- On the asymptotic maximal density of a set avoiding solutions to linear equations modulo a prime.With Pablo Candela. Acta Math. Hungar. 132 (3) (2011), 223–243.What is the largest density of a subset of Z/pZ that does not contain a solution to either of the equations x1 + x2 = x3, x1 + 2x2 = 4x3 + 5x4? This paper shows that if p is large then the answer to such questions is essentially given by an analogous quantity in R/Z, answering a Z/pZ-version of a question of Ruzsa about subsets of [N].
- A probabilistic technique for finding almost-periods of convolutions.With Ernie Croot. Geom. Funct. Anal. 20 (2010), 1367–1396.
- On the maximal number of three-term arithmetic progressions in subsets of Z/pZ.With Ben Green. Bull. London Math. Soc. 40 (2008), 945–955.
- A new proof of Roth's theorem on arithmetic progressions.With Ernie Croot. Proc. Amer. Math. Soc. 137 (2009), 805–809.
- Freiman isomorphisms between characters and linear limits of groups.We prove that the minimum number of 3APs in a subset of Z/pZ of density delta, divided by p2, is the same as the minimum amount of 3APs in a subset of the R/Z of density delta, up to o(1) errors as p tends to infinity. In fact, our results are rather more general than this, dealing with the general question of moving between linear equation counts over any compact abelian groups. E-mail me if interested!(still needs polishing) | brief
Notes- Bourgain's proof of the existence of long arithmetic progressions in A+B.These are some notes I wrote while trying to understand Bourgain's proof of the existence of long arithmetic progressions in sumsets A+B. I found it easiest to think of Bourgain's work as establishing an Lp-almost-periodicity result for convolutions of functions.
- A family of large density, large diameter sum-free sets in Z/pZ.Work of Deshouillers, Freiman and Lev has shown that large sum-free subsets of Z/pZ are necessarily somewhat structured, in the sense that they have a dilate contained in a short interval. In particular, this is known to hold for sets of density at least 0.318. In this note we construct a family of sum-free sets of density 0.25 that do not have a dilate contained in a short interval.(still needs polishing) | abstract
- An additive combinatorial take on Zeta constants.This is a short note demonstrating how one may interpret the values of Zeta(2), Zeta(4) and related sums in an additive combinatorial fashion. The basic idea is that one can view these values as representing the number of solutions to some linear equation in a simple subset of Z/pZ. What about Zeta(3)? Read the note and try it yourself!