# A Knapp example for Salem sets on the line

The restriction phenomenon in harmonic analysis is best known for surface measures on manifolds. A classical example is the unit sphere, where on the one hand we have the Stein-Tomas restriction theorem for $L^2$ densities on the sphere, and on the other hand, Stein’s restriction conjecture for $L^\infty$ densities remains open. (Partial intermediate results are also available, but that is a longer story that will have to wait for another time.)

However, restriction estimates can also be proved for fractal sets. This is due to Mockenhaupt (2000). Let $\mu$ be a compactly supported probability measure on $\mathbb{R}^n$ such that

$(1)\ \ \mu(B(x,r)) \leq C_1 r^{\alpha},\ \ x \in \mathbb{R}^n,\ r > 0$

$(2)\ \ |\widehat{\mu}(\xi)| \leq C_2 (1+|\xi|)^{-\beta/2},\ \xi\in\mathbb{R}^n$

Assume that the support of $\mu$ has Hausdorff dimension $0 < \alpha_0 < 1$. The second condition can be interpreted as a randomness condition: roughly, it says that $\mu$ has as little arithmetic structure as possible. It is well known that (1), (2) can only hold with $\alpha,\beta \leq \alpha_0$. We will say that $\mu$ is a Salem measure if $\alpha, \beta$ can both be taken arbitrarily close to $\alpha_0$.

For such measures, Mockenhaupt proved the following result, which we state here in dual form. Assume that

$(3)\ \ p > p_{n,\alpha,\beta} := \frac{2(2n-2\alpha+\beta)}{\beta}$,

then

$(4)\ \ \| \widehat{fd\mu} \|_{L^{p}(\mathbb{R}^n)} \leq C(p) \| f \|_{L^2(d\mu)}$

In the case of Salem measures on [0,1] such that (1) holds with $\alpha = \alpha_0$ and (2) holds for all $\beta$ less than $\alpha$, the range of $p$ in (3) is

$(5)\ \ p > (4 / \alpha ) -2$.

We are interested in the sharpness of the range of exponents in Mockenhaupt’s theorem for such measures. It’s relatively easy to see that $p$ cannot go below $2/ \alpha$. That, however, leaves an intermediate range of exponents for which the problem has been open.

Mockenhaupt’s proof follows the classical Tomas-Stein argument, and his range of exponents for $\alpha = \beta = n-1$ matches the Tomas-Stein range except for the endpoint. (An endpoint version of (4), with $p = p_{n,\alpha,\beta}$, was proved recently by Bak and Seeger. It is not known, however, whether it is even possible to have $\alpha = \beta = \alpha_0$ for fractal Salem measures. If we can only have $\beta$ strictly smaller than $\alpha$, then even the Bak-Seeger endpoint theorem only yields the strict inequality in (5), just for this reason.)

The Tomas-Stein exponent range for the sphere is known to be optimal. This can be seen by testing (4) on characteristic functions of small spherical caps (the Knapp example). Until now, there have been no similar constructions for fractal sets on the line. Mockenhaupt stated in his paper that he could not exclude the possibility that for Salem measures on [0,1] as above, (4) could in fact hold for $p>2/\alpha$ instead of (5). Bak and Seeger did not try to address this question. Based on conversations I’ve had at conferences, the conventional wisdom has been that (5) need not be optimal, given that there is no Knapp example in dimension 1.

Turns out, there is one.

Kyle Hambrook (my Ph.D. student) and I have just finished a paper in which we prove that the range of exponents in Mockenhaupt’s restriction theorem in the case of Salem sets in dimension 1 is sharp, except possibly for endpoint questions. Specifically, for any $\alpha = (\log t)/(\log n)$ with $t,n$ integer (a dense subset of (0,1)), and for all exponents $p$ with $p \text{\textless} (4 / \alpha ) -2$, we construct Salem sets of dimension $\alpha$ in $\mathbb{R}$ for which the restriction estimate fails.

The Salem sets $E$ are constructed via a randomized Cantor iteration similar to that in my earlier paper with Malabika Pramanik, but we modify the construction to make $E$ have large overlap with a nested sequence of small but arithmetically structured sets $F_l$. In a sense, this may be viewed as a one-dimensional analogue of Knapp’s counterexample. The latter is based on the fact that an almost flat spherical cap is contained in the curved sphere, or equivalently, that the sphere is tangent to a flat hyperplane. Here, the set $E$ may be thought of as random but nonetheless “tangent” to the arithmetically structured sets $F_\ell$. Our result then follows by testing the restriction estimate on the characteristic functions of $F_\ell$.

There remains the possibility that some Salem sets do not contain structured subsets, and that the range of exponents in the restriction estimate could be improved for such sets. However, our result shows that Mockenhaupt’s theorem in its stated generality (along with the Bak-Seeger endpoint estimate, if it turns out to be applicable to Salem sets) is optimal with regard to the range of exponents.

Here is the arXiv link. (Update: the paper is now published online in Geometric and Functional Analysis. The final publication is available here.)