
Recent Posts
 Multiple ergodic averages along functions from a Hardy field: convergence, recurrence and combinatorial applications
 Distal systems and Expansive systems
 Structure of multicorrelation sequences with integer part polynomial iterates along primes
 Tao’s Proof of (logarithmically averaged) Chowla’s conjecture for two point correlations
 A viewpoint on Katai’s orthogonality criterion
Category Archives: Number Theory
Tao’s Proof of (logarithmically averaged) Chowla’s conjecture for two point correlations
The Liouville function is the completely multiplicative function with for every prime . The Chowla conjecture predicts that this function behaves randomly. Here is a version of this conjecture. Conjecture 1 (Chowla) For every finite set , This conjecture received … Continue reading
Posted in Number Theory, Probability
Tagged Chowla, Entropy, Logarithmic Averages, primes, Tao
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A viewpoint on Katai’s orthogonality criterion
The Liouville function, defined as the completely multiplicative function which sends every prime to , encodes several important properties of the primes. For instance, the statement that is equivalent to the prime number theorem, while the improved (and essentially best … Continue reading
Posted in Classic results, Number Theory, Tool
Tagged Katai, Liouville function, primes, Sarnak, van der Corput
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Erdős Sumset conjecture
Hindman’s finite sums theorem is one of the most famous and useful theorems in Ramsey theory. It states that for any finite partition of the natural numbers, one of the cells of this partition contains an IPset, i.e., there exists … Continue reading
Posted in Combinatorics, Number Theory, State of the art
Tagged Banach density, Bohr sets, density, erdos, Jin, sumset, Ultrafilters, weak mixing
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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
Posted in Number Theory
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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
Posted in Combinatorics, Number Theory, Tool
Tagged erdos, multiplicative structure, Upper density, weighted densities
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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
Posted in Number Theory
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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
Posted in Classic results, Number Theory
Tagged Minkowski's theorem, polynomials, primes
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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
Posted in Number Theory
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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 semigroup. Let . We say that is (left) … Continue reading
Posted in Number Theory
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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 padic numbers being a particular case. I will introduce the idea behind this construction and explore … Continue reading
Posted in Number Theory
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