Tag - Arithmetic progressions

Yufei Zhao: Uniform Sets with Few Progressions via Colorings

9th May, 2023

Ruzsa asked whether there exist Fourier-uniform subsets of ℤ/Nℤ with very few 4-term arithmetic progressions (4-AP). The standard pedagogical example of a Fourier-uniform set with a "wrong" density of 4-APs actually has 4-AP density much higher than random. Can it instead be much lower than random? Gowers constructed Fourier uniform sets with 4-AP density at most α4+c. It remains open whether a superpolynomial decay is possible. We will discuss this question and some variants. We relate it to an arithmetic Ramsey question: can one No(1)-color of [N] avoiding symmetrically-colored 4-APs?

Xuancheng Shao: Gowers uniformity of primes in arithmetic progressions

27th July, 2020

A celebrated theorem of Green-Tao asserts that the set of primes is Gowers uniform, allowing them to count asymptotically the number of k-term arithmetic progressions in primes up to a threshold. In this talk I will discuss results of this type for primes restricted to arithmetic progressions. These can be viewed as generalizations of the classical Bombieri-Vinogradov theorem. I will also discuss a number of applications; for example, the set of primes p obeying explicit bounded gaps.

Thomas Bloom: Breaking the logarithmic barrier in Roth’s theorem on progressions

10th July, 2020

(joint work with Olof Sisask) We present an improvement to Roth's theorem on arithmetic progressions, by showing that if A ⊂ [N] has no non-trivial three-term arithmetic progressions then |A| ≪ N/(log N)1+c for some positive absolute constant c. In particular, this establishes the first non-trivial case of a conjecture of Erdős on arithmetic progressions.

Tomasz Schoen: Improved bound in Roth’s theorem

22nd May, 2020

I sketch a proof of a new bound in Roth's theorem on arithmetic progressions: if A ⊆ {1,...,N} does not contain any non-trivial three-term arithmetic progression then |A| ≪ (log log N)^3+o(1)N/log N.

Yufei Zhao: Popular common difference

19th May, 2020

Green proved the following strengthening of Roth's theorem: for every positive ϵ, there is some n(ϵ) such that for every N ≥ n(ϵ) and A ⊂ [N] with |A| = αN, there is some nonzero d such that A contains at least (α3 − ϵ)N three-term arithmetic progressions with common difference d (i.e., a popular common difference with frequently at least roughly the random bound). I'll discuss some extensions and generalizations of this result:

   •  How large does n(ϵ) have to be for the result to hold? (It turns out that a tower-type bound is necessary)
   •  Besides 3-term arithmetic progressions, is there a similar result for other patterns?
   •  What about patterns in higher-dimensional patterns?