Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Falconer-type estimates for dot products   We present a family of sharpness examples for Falconer-type single dot
product results. In particular, for $d\geq 2,$ for any $s<\frac{d+1}{2},$ we
construct a Borel probability measure $\mu$ satisfying the energy estimate
$I_s(\mu)<\infty,$ yet the estimate \begin{equation} (\mu \times
\mu)\{(x,y):1\leq x\cdot y \leq 1+\epsilon\} \leq C\epsilon \end{equation} does
not hold with constants independent of $\epsilon$. It is known (\cite{EIT11})
that such an estimate always holds with $C$ independent of $\epsilon$ if
$I_{\frac{d+1}{2}}(\mu)<\infty$. Thus our estimate proves the sharpness of the
dimensional threshold in this result and generalizes similar results
(\cite{Mat95}, \cite{IS16}) established in the case when the dot product $x
\cdot y$ is replaced by the Euclidean distance function $|x-y|$, or, more
generally, ${||x-y||}_K$, the distance that comes from the norm induced by a
symmetric convex body $K$ with a smooth boundary and non-vanishing curvature.
Our constructions are partially based on ideas that come from discrete
incidence theory.

25
The categories of this paper are: patt-sol
6
The categories of this paper are: astro-ph
35
The categories of this paper are: mtrl-th
17
The categories of this paper are: q-fin
15
The categories of this paper are: nucl-ex
8
The categories of this paper are: hep-ex
0
The categories of this paper are: hep-ph
3
The categories of this paper are: physics
4
The categories of this paper are: cond-mat
29
The categories of this paper are: funct-an
36
The categories of this paper are: cmp-lg
10
The categories of this paper are: q-bio
12
The categories of this paper are: nucl-th
14
The categories of this paper are: math-ph
26
The categories of this paper are: bayes-an
21
The categories of this paper are: adap-org