By Pertti Mattila
In past times 20 years there was energetic interaction among geometric degree idea and Fourier research. This publication describes a part of that improvement, focusing on the connection among the Fourier rework and Hausdorff measurement. the most subject matters crisis functions of the Fourier remodel to geometric difficulties regarding Hausdorff size, reminiscent of Marstrand variety projection theorems and Falconer's distance set challenge, and the function of Hausdorff measurement in glossy Fourier research, in particular in Kakeya equipment and Fourier restrict phenomena. The dialogue contains either classical effects and up to date advancements within the region. the writer emphasises partial result of very important open difficulties, for instance, Falconer's distance set conjecture, the Kakeya conjecture and the Fourier restrict conjecture. basically self-contained, this publication is acceptable for graduate scholars and researchers in arithmetic
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Extra info for Fourier analysis and Hausdorff dimension
These facts are a reflection of the Heisenberg uncertainty principle: a function and its Fourier transform cannot both be small. For more on this, see Havin and J¨oricke  and Wolff . 2 The fact that every Schwartz function is a Fourier transform of another Schwartz function is very useful for construction of various examples with desired properties. For example, we can find a non-negative function ϕ 2 S(Rn ) such that ϕ 1 on B(0, 1), ϕ 0 and spt ϕ B(0, 1) (or vice versa, ϕ 1 on B(0, 1) and spt ϕ B(0, 1)).
53) then for R > 1, jμj2 Rn s . 2 choose ϕ so that ϕ 0, ϕ 1 on B(0, 1) and spt ϕ B(0, 1), and observe that then 2 ϕR jμj2 and ϕR μ(x) R n s . 53) hold, we have B(0,R) jμj2 R n s for R > 1. Strichartz  and [1990a] made a much more detailed study of such ball averages and related matters. 1 jf μj2 D c(n, s) n jf j2 dμ, B(0,R) for some positive and finite constant c(n, s). To get an idea when f D 1, notice that if ϕ approximates well the characteristic function of B(0, 1) and ϕR is as above, then R s n B(0,R) jf μj2 is close to R s n ϕR μ dμ by the above arguments, and the convergence of r s μ(B(x, r)) as r !
In particular, Chen constructs measures as in Mitsis’s question, except that he needs a logarithmic factor in one of the conditions. Related results can also be found in K¨orner  and Shmerkin and Suomala . From the above we know that if a set has zero s-dimensional Hausdorff measure, then it cannot support a non-trivial measure whose Fourier transform would tend to zero at infinity faster than jxj s/2 . But how quickly can they tend to zero in terms of ϕ(jxj) for various functions ϕ?