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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 [1995] and Wolff [2003]. 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 [1989] 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 [2011] and Shmerkin and Suomala [2014]. 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?