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Miguel N. Walsh

Publications and source records attributed to Miguel N. Walsh.

9 recordsLinked to original sources
arXiv2023

Stability under scaling in the local phases of multiplicative functions

We introduce a strategy to tackle some known obstructions of current approaches to the Fourier uniformity conjecture. Assuming GRH, we then show the conjecture holds for intervals of length at least $(\log X)^{ψ(X)}$, with $ψ(X) \rightarrow \infty$ an arbitrarily slowly growing function of $X$. We expect the methods should adapt to nilsequences, thus also showing that the Generalised Riemann Hypothesis implies close to exponential growth in the sign patterns of the Liouville function.

math.NT↗
arXiv2023

Phase relations and pyramids

We develop tools to study the averaged Fourier uniformity conjecture and extend its known range of validity to intervals of length at least $\exp(C (\log X)^{1/2} (\log \log X)^{1/2})$.

math.NT↗
arXiv2021

Local uniformity through larger scales

By associating frequencies to larger scales, we provide a simpler way to derive local uniformity of multiplicative functions on average from the results of Matomäki-Radziwill.

math.NT↗
arXiv2020

The polynomial method over varieties

We establish sharp estimates that adapt the polynomial method to arbitrary varieties. These include a partitioning theorem, estimates on polynomials vanishing on fixed sets and bounds for the number of connected components of real algebraic varieties. As a first application, we provide a general incidence estimate that is tight in its dependence on the size, degree and dimension of the varieties involved.

math.AG↗
arXiv2019

Concentration estimates for algebraic intersections

We present an approach over arbitrary fields to bound the degree of intersection of families of varieties in terms of how these concentrate on algebraic sets of smaller codimension. This provides in particular a substantial extension of the method of degree-reduction that enables it to deal efficiently with higher-dimensional problems and also with high-degree varieties. We obtain sharp bounds that are new even in the case of lines in $\mathbb{R}^n$ and show that besides doubly-ruled varieties, only a certain rare family of varieties can be relevant for the study of incidence questions.

math.AG↗
arXiv2013

The algebraicity of ill-distributed sets

We show that every set S in [N]^d occupying less than p^t residue classes for some real number t < d and every prime p, must essentially lie in the solution set of a polynomial equation of degree at most (log N)^C, for some constant C depending only on t and d. This provides the first structural result for arbitrary t < d and S.

math.NT↗
arXiv2013

Bounded rational points on curves

We establish the sharp estimate <<_d N^{2/d} for the number of rational points of height at most N on an irreducible projective curve of degree d. We deduce this from a result for general hypersurfaces that is sensitive to the coefficients of the corresponding form.

math.NT↗
arXiv2011

The inverse sieve problem in high dimensions

We show that if a big set of integer points in [0,N]^d, d>1, occupies few residue classes mod p for many primes p, then it must essentially lie in the solution set of some polynomial equation of low degree. This answers a question of Helfgott and Venkatesh.

math.NT↗