Who?

I have moved to Uppsala University, but this page remains somewhat active. Before the move, I was a Researcher at the Department of Mathematics at KTH, the Royal Institute of Technology. My research interests generally lie in probability, combinatorics, harmonic analysis and number theory. Know me from somewhere? I have previously been stationed at the University of Cambridge, MIT, University of Bristol, Queen Mary, University of London, University of British Columbia, and the Simons Institute for the Theory of Computing.

Google Scholar provides some interesting links to papers and articles surrounding my research.

Want to study the emergence of a giant puddle in the random raindrop model? Here's a visualisation. (Puddles merge according to their barycentres.)

Contact details

You can usually reach me at olof.sisask[AT-tecken]math.uu.se.

Public engagement

The interactive tools we used during my public lectures at Alba Nova on pi-day 2018 and at Senioruniversitetet in Dec 2018 can be found at:
We also used these during the highschool teacher development day Kleinerdagen at the Royal Swedish Academy of Sciences (KVA) on 31 Oct 2017. If you attended these and want any of the other materials we used, please email me!

Teaching

I am currently lecturing Calculus; if you are a student on this course you are of course welcome to contact me with any questions you might have.

An extract of the teaching evaluation for Discrete Mathematics at KTH.

Some notes:

- Discrete Fourier analysis: structures in sumsets (course at the CIMPA Research School on
*Fourier analysis of groups in combinatorics*in Shillong, India, 2013)

Papers and articles

- Logarithmic bounds for Roth's theorem via almost-periodicity.With Thomas Bloom. Submitted.
- Convolutions of sets with bounded VC-dimension are uniformly continuous.Preprint.
- Roth's theorem for four variables and additive structures in sums of sparse sets.With Tomasz Schoen. Forum of Mathematics, Sigma 4 (2016) e5 (28 pages).
- Convergence results for systems of linear forms on cyclic groups, and periodic nilsequences.With Pablo Candela. SIAM J. Discrete Math. 28 (2) (2014), 786–810.
- Arithmetic progressions in sumsets andWith Ernie Croot and Izabella Laba. Combin. Probab. Comput. 22 (3) (2013), 351–365.
`L`^{p}-almost-periodicity. - A removal lemma for linear configurations in subsets of the circle.With Pablo Candela. Proc. Edinburgh Math. Soc. 56 (3) (2013), 657–666.
- 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**/`p`**Z**that does not contain a solution to either of the equations`x`+_{1}`x`=_{2}`x`,_{3}`x`+ 2_{1}`x`= 4_{2}`x`+ 5_{3}`x`? This paper shows that if_{4}`p`is large then the answer to such questions is essentially given by an analogous quantity in**R**/**Z**, answering a**Z**/`p`**Z**-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 ofWith Ben Green. Bull. London Math. Soc. 40 (2008), 945–955.
**Z**/`p`**Z**. - A new proof of Roth's theorem on arithmetic progressions.With Ernie Croot. Proc. Amer. Math. Soc. 137 (2009), 805–809.
- A family of large density, large diameter sum-free sets in
**Z**/`p`**Z**.Work of Deshouillers, Freiman and Lev has shown that large sum-free subsets of**Z**/`p`**Z**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 - Freiman isomorphisms between characters and linear limits of groups.We prove that the minimum number of 3APs in a subset of
**Z**/`p`**Z**of density delta, divided by`p`^{2}, 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`L`^{p}-almost-periodicity result for convolutions of functions. - 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**/`p`**Z**. What about Zeta(3)? Read the note and try it yourself!