Tag - Additive number theory

Kevin Ford: Prime gaps, probabilistic models, the interval sieve, Hardy-Littlewood conjectures and Siegel zeros

3rd September, 2020

Motivated by a new probabilistic interpretation of the Hardy-Littlewood k-tuples conjectures, we introduce a new probabilistic model of the primes and make a new conjecture about the largest gaps between the primes below x. Our bound depends on a property of the interval sieve which is not well understood. We also show that any sequence of integers which satisfies a sufficiently uniform version of the Hardy-Littlewood conjectures must have large gaps of a specific size. Finally, assuming that Siegel zeros exist we show the existence of gaps between primes which are substantially larger than the gaps which are known unconditionally. Much of this work is joint with Bill Banks and Terry Tao.

Will Sawin: The quadratic Bateman-Horn conjecture over 𝔽q[u]

17th August, 2020

The Bateman-Horn conjecture predicts the fraction of integers n such that n2+1 is prime, and makes similar predictions for polynomials of higher degree. In joint work in progress with Mark Shusterman, we prove an analogue of the n2+1 case, replacing natural numbers n with polynomials in 𝔽q[u], which for instance counts the fraction of polynomials f such that f2+u is an irreducible polynomial. The proof combines geometric methods, unusual algebraic properties of polynomials, and some (very) classical number theory.

Ofir Gorodetsky: The distribution of squarefree integers in short intervals

11th August, 2020

The squarefree integers are divisible by no square of a prime. It is well known that they have a positive density within the integers. We consider the number of squarefree integers in a random interval of size H : |{n ∈ [x,x+H] : n is squarefree}|, where x is a random number between 1 and X. The variance of this quantity has been studied by R. R. Hall in 1982, obtaining asymptotics in the range H less than X2/9, with a proof method that stays in 'physical space'. Keating and Rudnick recently conjectured that his result persists for the entire range H less than X1-ε. We make progress on this conjecture, with properties of the Riemann zeta function playing a role in our results. We will show how, on RH, one can verify the conjecture for H up to X2/3.

Akshat Mudgal: Diameter free estimates for Vinogradov systems

10th August, 2020

A classical object of study in additive number theory has been the Vinogradov system, that is, the system defined by the equations x1j+ . . . + xsj = y1j+ . . . + ysj (j = 1, . . ., k). In particular, given a finite set A of integers, finding sharp upper bounds for the number of solutions Js,k(A) to this system, when all the variables lie in the set A, has been an important topic of work. Recently, two major approaches have been developed to tackle this problem - the efficient congruencing method of Wooley, and the decoupling techniques of Bourgain-Demeter-Guth. Both these methods give upper bounds for Js,k(A) in terms of s,k, and the cardinality |A| of A, and the diameter X of A. In particular, when X is large in terms of |A|, say when X is much larger than exp(exp(|A|)), these bounds perform worse than the trivial estimates. In this talk, we present new upper bounds for Js,2(A) which depend only on |A| and s. These improve upon, and generalize, a previous result of Bourgain and Demeter.

Julian Sahasrabudhe: Zeros of cosine polynomials, a problem of Littlewood

4th August, 2020

Let f be a {0,1}-cosine polynomial with n terms. In his 1986 monograph, J.E. Littlewood considered the minimum number of zeros that such polynomials have in [0,2π] and conjectured that the number of such roots is "n-1 or not much less". While it is now known that there exist cosine polynomials with considerably fewer roots, much less is known about the lower bound and, in fact, it was a long standing problem just to show that the number of such zeros tends to infinity with n. We will discuss the resolution of this conjecture and also mention some more recent progress on the upper bound.

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.

Nikos Frantzikinakis: Ergodic properties of the Liouville function and applications

20th July, 2020

The Liouville function is a multiplicative function that encodes important information related to distributional properties of the prime numbers. A conjecture of Chowla states that the values of the Liouville function fluctuate between plus and minus in such a random way, that all sign patterns of a given length appear with the same frequency. The Chowla conjecture remains largely open and in this talk we will see how ergodic theory combined with some feedback from number theory allows us to establish two variants of this conjecture. Key to our approach is an in-depth study of measure preserving systems that are naturally associated with the Liouville function. The talk is based on joint work with Bernard Host.

Sarah Peluse: An asymptotic version of the prime power conjecture for perfect difference sets

13th July, 2020

A subset D of a finite cyclic group ℤ/mℤ is called a "perfect difference set" if every nonzero element of ℤ/mℤ can be written uniquely as the difference of two elements of D. If such a set exists, then a simple counting argument shows that m=n2+n+1 for some nonnegative integer n. Singer constructed examples of perfect difference sets in ℤ/(n2+n+1)ℤ whenever n is a prime power, and it is an old conjecture that these are the only such n for which a perfect difference set exists. In this talk, I will discuss a proof of an asymptotic version of this conjecture: the number of n less than N for which ℤ/(n2+n+1)ℤ contains a perfect difference set is asymptotically the number of prime powers less than N.

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.

Sophie Stevens: An update on the sum-product problem

6th July, 2020

In new work with Misha Rudnev, we prove a stronger bound on the sum-product problem, showing that

max(|A + A|, |AA|) ≥ |A|4/3 + 2/1167 − o(1)

for any finite set A of real numbers. This builds upon the work of Solymosi, Konyagin and Shkredov, although our paper is self-contained. I will give an overview of the arguments, both old and new, and describe some consequences of the new arguments.