Polynomial configurations in fractal sets: Kevin Henriot, Malabika Pramanik and I have posted a paper where we prove the following result: if a measure μ on a fractal set E in Rn has Fourier decay with some exponent β, and if it also obeys a ball condition with exponent α close enough to n (depending on β and on the constants in both conditions), then it must contain nontrivial configurations given by certain types of systems of matrices with a polynomial term. This is somewhat similar to my earlier paper with Vincent Chan and Malabika Pramanik, on configurations given by systems of linear forms, but there are significant differences. One is of course the polynomial term: we use stationary phase estimates to control the corresponding part of the “counting form” Λ. (Interestingly, while said stationary estimates apply to functions much more general than polynomials, the polynomial form of the nonlinear term is required for the “continuous” estimates which are based on a number-theoretic argument.) Another is that any rate of Fourier decay β>0 suffices, with the caveat that α must be close enough to n, where “close enough” now depends on both the constants and β. This improvement is due to more efficient use of restriction estimates, and extends to the result with Chan and Pramanik as well as my earlier paper with Pramanik on 3-term arithmetic progressions in fractals. A recent result of Pablo Shmerkin shows that the dependence on constants cannot be removed: he proves, for example, that there exists a 1-dimensional (but of Lebesgue measure 0) Salem set on the line that does not contain a nontrivial 3-term arithmetic progression.
Fractal Knapp examples: Kyle Hambrook and I have been asked on various occasions whether our “Knapp example” for fractal sets on the line could be extended to fractals in higher dimensions. In this paper, we combined our construction (with modifications due to Chen) and the classical Knapp example on the sphere to produce fractal Knapp examples of dimension between n-1 and n in Rn.
My profile for Women in Maths: this was published a while ago, in case anyone here is interested.
It’s been a while since I posted any photos here, so here’s one I took today. There will be more on my Google+ page.