6 citations · 8 across the 5 of their papers we have counts for
9 papers
Stability under scaling in the local phases of multiplicative functions
Miguel N. Walsh
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…
Phase relations and pyramids
Miguel N. Walsh
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/…
Local uniformity through larger scales
Miguel N. Walsh
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.
Concentration estimates for algebraic intersections
Miguel N. Walsh
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…
The polynomial method over varieties
Miguel N. Walsh
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 bou…
Bounded rational points on curves
Miguel N. Walsh
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…
The algebraicity of ill-distributed sets
Miguel N. Walsh
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 equ…
Norm convergence of nilpotent ergodic averages
Miguel N. Walsh
We show that multiple polynomial ergodic averages arising from nilpotent groups of measure preserving transformations of a probability space always converge in the L^2 norm.
The inverse sieve problem in high dimensions
Miguel N. Walsh
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 polynomi…