Category Archives: Number Theory

Alternative proofs of two classical lemmas

Two of the most fundamental tools in ergodic Ramsey theory are the mean ergodic theorem and the van der Corput trick. Both have a classical and fairly simple proof, which I have presented before in the blog. Recently I came … Continue reading

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Weighted densities with multiplicative structure

The upper density of a set , defined by provides a useful way to measure subsets of . For instance, whenever , contains arbitrarily long arithmetic progressions, this is Szemerédi’s theorem. A fundamental property of the upper density is that … Continue reading

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Description of my work using few words

There is a famous xkcd describing a space ship using only the one thousand most used words. Since then some people have tried to describe their work using only those thousand words, and even a quite useful text editor was built. I … Continue reading

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Primes of the form x^2+2y^2

In this post I will present a quite nice proof of the following fact from elementary number theory: Theorem 1 Let be a prime number. There are such that if and only if is a quadratic residue . Recall that … Continue reading

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Applications of the coloring trick

— 1. Introduction — In a previous post I used a coloring trick of Bergelson to deduce Brauer’s theorem from Szemeredi’s theorem. In this post I will provide two more applications of the coloring trick. To put it simply, the … Continue reading

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Multiplicative obstructions to syndeticity on the Natural numbers

The notion of syndetic set has some important applications. For instance, the definition, due to Bohr, of almost periodic functions relies on syndetic sets. Definition 1 (Syndetic set) Let be a topological semi-group. Let . We say that is (left) … Continue reading

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Absolute values on Number Fields

One fundamental construction in modern algebraic number theory is that of the completion of a number field with respect to an absolute value, the p-adic numbers being a particular case. I will introduce the idea behind this construction and explore … Continue reading

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