A probabilistic technique for finding almost-periods of convolutions

We introduce a new probabilistic technique for finding 'almost-periods' of convolutions of subsets of groups. This gives results similar to the Bogolyubov-type estimates established by Fourier analysis on abelian groups but without the need for a nice Fourier transform to exist. We also present applications, some of which are new even in the abelian setting. These include a probabilistic proof of Roth's theorem on three-term arithmetic progressions and a proof of a variant of the Bourgain-Green theorem on the existence of long arithmetic progressions in sumsets A+B that works with sparser subsets of {1, ..., N} than previously possible. In the non-abelian setting we exhibit analogues of the Bogolyubov-Freiman-Halberstam-Ruzsa-type results of additive combinatorics, showing that product sets A B C and A^2 A^{-2} are rather structured, in the sense that they contain very large iterated product sets. This is particularly so when the sets in question satisfy small-doubling conditions or high multiplicative energy conditions. We also present results on structures in product sets A B. Our results are 'local' in nature, meaning that it is not necessary for the sets under consideration to be dense in the ambient group. In particular, our results apply to finite subsets of infinite groups provided they 'interact nicely' with some other set.Comment: 29 pages, to appear in GAF

Roth's theorem for four variables and additive structures in sums of sparse sets

We show that if a subset A of {1,...,N} does not contain any solutions to the equation x+y+z=3w with the variables not all equal, then A has size at most exp(-c(log N)^{1/7}) N, where c > 0 is some absolute constant. In view of Behrend's construction, this bound is of the right shape: the exponent 1/7 cannot be replaced by any constant larger than 1/2. We also establish a related result, which says that sumsets A+A+A contain long arithmetic progressions if A is a subset of {1,...,N}, or high-dimensional subspaces if A is a subset of a vector space over a finite field, even if A has density of the shape above.Comment: 23 page

On the maximal number of three-term arithmetic progressions in subsets of Z/pZ

Let a be a real number between 0 and 1. Ernie Croot showed that the quantity \max_A #(3-term arithmetic progressions in A)/p^2, where A ranges over all subsets of Z/pZ of size at most a*p, tends to a limit as p tends to infinity through primes. Writing c(a) for this limit, we show that c(a) = a^2/2 provided that a is smaller than some absolute constant. In fact we prove rather more, establishing a structure theorem for sets having the maximal number of 3-term progressions amongst all subsets of Z/pZ of cardinality m, provided that m < c*p.Comment: 12 page

Professor Dragendorffi teadustegevuse jäljed Tartu Ülikooli ajaloo muuseumi kogudes

Evidence of Dragendorff’s scientific work inThe University of Tartu History MuseumCollectionsJ. G. N. Dragendorff (20.04.1836 – 26.03.1898) was a Professor of Pharmacy at the University of Tartu from 1864 to 1894. The development of equipment and increase in the number of chemical and herbal substances used (3882 different substances) was significant between 1840 and 1885. Currently, there are 5 different collections of herbal and chemical substances in the University of Tartu HistoryMuseum from the Dragendorff period: Herbal substances, Microscopic preparations, Park substances, Wallpaper sample and Textile sample collections. The collection of herbal substances is the largest (ca 1200). Mostherbal substance exhibits are from the period between 1879 and 1889. However, there are also some exhibits from earlier times (for example 1829 Huanuco, 1846 Hamburg, 1863 Paris and 1867 Venezuela). Professor Dragendorff conducted extensive research to discover the cure for malaria – he was sent Cinchona (Cinchonae cortec) from overseas to be able to prepare a pharmaceutical that helped cure malaria. The microscopic preparations are also from late 19th century – all together 1197 preparations have been described.The Park substances collection consists of 90 closed glassampoules. Research has focused on park substances extracted from 7 different herbal substances or plants. Wallpaper sample and textile sample collections demonstrate the research focused on finding arsenic in wallpaper and textiles to possibly explain some cases of poisoning. The collection consists of approximately 700 wallpaper and 200 textile samples

Eesti muuseumide meditsiinilooliste kogude ja esemete väljaselgitamine

Mapping of the Collections of History of Medicinein Estonian MuseumsLeili Kriis, Sirje SisaskUniversity of Tartu MuseumThe work plan of the University of Tartu Museum for the followingyears includes creating a new exposition for the university’s Old AnatomicalTheatre. This made us wonder, what kinds of collections ofhistory of medicine are there in other Estonian museums. A greatopportunity to obtain information opened up with the Ministry ofCulture’s support programme for developing museums that we participatedin with our project “Mapping of the Collections of History ofMedicine in Estonian Museums” from 17 April to 31 December 2013.To obtain data, we designed a questionnaire for the chief treasurersof the museums. Based on the answers we received, the followingoverview was compiled.At the time we conducted the survey, more than half of the materialsof history of medicine in the museum collections had beenentered also into the Estonian Museum Information System MuIS.Keywords related to the topic of medicine have been used. Most ofthe corresponding materials in Estonian museums are connected topharmacies (material objects as well as photographs and archive materials),and furnishings of dental practices (Pärnu Museum, VõruCounty Museum, etc.). Regional museums also contain objects of veterinarymedicine. There are rare items of history of medicine suchas an inoculation knife, cupping lancets, bloodletting devices, a bonesaw, an irritation instrument, healing stones, a homoeopathic pharmacy,and many more. The most noteworthy archive materials areprobably the inoculation materials, mud treatment materials (materialsand manuscripts of Dr Hunnius from Haapsalu, mud treatmenthistory of Saaremaa), and manuscripts by medics (historical overviews).Topics like doping (the Estonian Sports Museum), designingmedical institutions (hospitals, sanatoriums, etc.) (Museum of EstonianArchitecture, Museum of Viljandi), diseases and stress, and howthese are reflected in a person’s creative work (Estonian Theatre andMusic Museum) have also been documented. Most valuable are thestories found in the collections of county museums that tell of medicalinstitutions (hospitals, pharmacies, sanatoriums, leprosariums, etc.)that have operated or are operating in the region and reflect the specificcharacter of the region, as well as the materials of well-known localmedics. Also materials related to alternative and folk medicine. Allthis enriches the general picture of the Estonian history of medicine.Our working group reached the conclusion that the contents ofthe MuIS dictionary should be structured in a better manner andsupplemented with topics that are reflected in specific museum collections(alternative medicine, nutrition, environment, etc.). Imagesof museum objects are also important data carriers in the database.Rare historical documents deserve to be digitised as soon as possiblesince they tend to become fragile with time. Regional museumsshould also find room for materials related to the history and doctorsof local medical institutions