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\title[Fractal and multifractal Fourier series]{\bf Fractal and multifractal Fourier series}


\author[F. Chamizo]{Fernando Chamizo}
\date{\footnotesize Hauteurs, Modularit\'e et Transcendance}
\institute{{\small Autonomous University of Madrid} \\[6pt] {\footnotesize \url{http://www.uam.es/fernando.chamizo}}
\\
\
\\
{\footnotesize Marseille. May  15th 2014}}

\begin{document}

\setbeamertemplate{headline}
{%
\begin{beamercolorbox}{section in head/foot}
%\alert{\ \insertframenumber\ de \inserttotalframenumber}
\ \vskip6.5pt\par \insertnavigation{\paperwidth}\vskip2pt
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}

%%%%%%%%%%
% TITLE %%
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\begin{frame}
\titlepage
\end{frame}



%%%%%%%%%%
% SLIDE %%
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\begin{frame}
\frametitle{Credits}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
This talk is based mainly on the joint work with A. Ubis:

\begin{itemize}
 \item  \color{blue}
Multifractal behavior of polynomial Fourier series. {\it Adv. Math.} 250 (2014), 1--34.
\end{itemize}

\end{block}}


{\setbeamercolor{block title}{bg=white}
\begin{block}{}
There is also a (unpublished) part coauthored with S. Ruiz-Cabello and included in his recent PhD thesis to be defended in two weeks:

\begin{itemize}
 \item \color{blue}
Prime generators,  approximate identities and multifractal functions. PhD Thesis 2014.
\end{itemize}

\end{block}}


\end{frame}



%%%%%%%%%%
% INDEX %%
%%%%%%%%%%
%\begin{frame}
%\frametitle{Table of contents}
%%\tableofcontents[pausesections]
%\tableofcontents
%\end{frame}




%%%%%%%%%%
% SLIDE %%
%%%%%%%%%%
\section{Fractal and multifractal}
\begin{frame}
\frametitle{Fractal and multifractal functions}



\vspace{-0.75cm}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
A way of constructing fractals is to consider low regularity functions.
\end{block}}

\vspace{-0.4cm}

\begin{center}
\begin{tabular}{cc}
\begin{tabular}{c}
\includegraphics[scale=0.3]{./images/weier.png}
\end{tabular}
&
\hspace{-10pt}\begin{tabular}{l}
Weierstrass function
\\[15pt]
$\displaystyle\sum_{n=0}^\infty\frac{\sin(2\pi 4^nx)}{2^n}$
\\[15pt]
%$\beta_f(x)=1/2$
\end{tabular}
\end{tabular}
\end{center}


\vspace{-0.5cm}


\[
 \beta_f(x)=\sup\big\{\gamma\le 1\; :\; |f(x+h)-f(x)|=O(|h|^\gamma)\big\}
\]

\begin{footnotesize}
\color{gray}
[Definition to be modified at points of differentiability]
\end{footnotesize}

\end{frame}


%%%%%%%%%%
% SLIDE %%
%%%%%%%%%%
\begin{frame}


\[
 \beta_f(x)=\sup\big\{\gamma\le 1\; :\; |f(x+h)-f(x)|=O(|h|^\gamma)\big\}
\]


\begin{center}
\begin{tabular}{cc}
\begin{tabular}{c}
\includegraphics[scale=0.3]{./images/weier.png}
\end{tabular}
&
\hspace{-10pt}\begin{tabular}{l}
Weierstrass function
\\[15pt]
$\displaystyle\sum_{n=0}^\infty\frac{\sin(2\pi 4^nx)}{2^n}$
\\[15pt]
$\beta_f(x)=1/2$
\end{tabular}
\end{tabular}
\end{center}

\vspace{-15pt}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
A function with $\beta_f(x)=\beta$ (constant) gives a fractal graph of dimension $2-\beta$.
\end{block}}


{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\small\color{gray}\bf
For Weierstrass function one could suspect that $\beta_f(x)=1/2$ because it has almost $1/2$ of derivative.
\end{block}}




\end{frame}




%%%%%%%%%%
% SLIDE %%
%%%%%%%%%%
\begin{frame}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
A \emph{multifractal} set is a set made of infinitely many interwoven fractal subsets of different dimensions. We can get one of these objects when the \emph{H\"older exponent} $\beta_f(x)$ is constant on fractal sets. 
\end{block}}

\pause

\begin{block}{Spectrum of singularities}
\[
 d(\beta)=\dim \{x\;:\; \beta_f(x)=\beta\}
\]
where $\dim$ is the Hausdorff dimension {\footnotesize ($\dim\emptyset =-\infty$ or undefined)}.
\end{block}

\begin{block}{Multifractal function}
It is a function  with a non-discrete spectrum of singularities.
\end{block}


\begin{center}
\begin{tabular}{cc}
\begin{tabular}{c}
$\scriptstyle d(\beta)$
\\
{\tiny Pure fractal}
\end{tabular}
\hspace{-9pt}
\begin{tabular}{c}
\includegraphics[scale=0.15]{./images/puref.png}
\end{tabular}
&
\begin{tabular}{c}
$\scriptstyle d(\beta)$
\\
{\tiny Multifractal}
\end{tabular}
\hspace{-9pt}
\begin{tabular}{c}
\includegraphics[scale=0.15]{./images/multif.png}
\end{tabular}
\end{tabular}
\end{center}


\end{frame}


%%%%%%%%%%
% SLIDE %%
%%%%%%%%%%
\begin{frame}


{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\textbf{Why should I care about multifractals?} 
Multifractal analysis is employed in many areas, ranging from fluid mechanics to medical imaging.
\end{block}}

\pause

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\textbf{Why should I care if I am attending this conference?} 
The study of the spectrum of singularities of simple functions leads to highly nontrivial questions about Diophantine approximation.
\end{block}}

\pause

{\setbeamercolor{block title}{bg=pink}
\begin{block}{}
In this talk the \emph{simple function} is mainly a Fourier series with integral polynomial frequencies
\[
 F(x)=\sum_{n=1}^\infty\frac{e\big(P(n)x\big)}{n^{\alpha}},
\qquad P\in\Z[x]
\quad\text{ where }e(x)=e^{2\pi i x}.
\]
\end{block}}


\end{frame}


%%%%%%%%%%
% SLIDE %%
%%%%%%%%%%
\begin{frame}




{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\[
 F(x)=\sum_{n=1}^\infty\frac{e\big(P(n)x\big)}{n^{\alpha}},
\qquad P\in\Z[x]
\quad\text{ where }e(x)=e^{2\pi i x}.
\]
\end{block}}




{\setbeamercolor{block title}{bg=green}
\begin{block}{\color{black}}
\textbf{The question:} 
What is the spectrum of singularities of $F$?
\end{block}}

\pause

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\textbf{Short answer:} 
I don't know.
\end{block}}

\pause

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\textbf{Long answer:} 
The case $\deg P=2$ is settled, there is a general lower bound, and also an upper bound proving that $F$ is a multifractal function in certain ranges, there are some conjectural formulas and heuristics{\dots}   [The rest of the talk]. 
\end{block}}


\end{frame}



%%%%%%%%%%
% SLIDE %%
%%%%%%%%%%
\section{Quadratic case}
\begin{frame}
\frametitle{What is special in the quadratic case?}


{\setbeamercolor{block title}{bg=white}
\begin{block}{}
The simplest quadratic example is indeed old and famous. 
\end{block}}


\begin{block}{Riemann's example}
\[
 R(x)=\sum_{n=1}^\infty
\frac{\sin(2\pi n^2x)}{n^2}
\]
\end{block}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
According to Weierstrass (1861), Riemann
considered $R$ to be an example of a continuous nowhere differentiable function. 
\end{block}}


\end{frame}


%%%%%%%%%%
% SLIDE %%
%%%%%%%%%%
\begin{frame}



{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\begin{center}
\begin{tabular}{c}
\includegraphics[scale=0.481]{./images/r_g.png}
\\
\color{blue}Graph of the Riemann example
\end{tabular}
\end{center}
\end{block}}



\end{frame}



%%%%%%%%%%
% SLIDE %%
%%%%%%%%%%
\begin{frame}

\[
 R(x)=\sum_{n=1}^\infty
\frac{\sin(2\pi n^2x)}{n^2}
\]

\vspace{-15pt}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
G.H. Hardy (1916): $R$ is not differentiable at any irrational value and
at some families of rational values. 
J.~Gerver (1970): $R$ is differentiable at the rationals not considered by Hardy.
\end{block}}

\vspace{-6pt}

\uncover<3->{\[
\theta(z)=\sum_{n\in\Z} e(n^2z),
\qquad \Im z>0
\]}

\vspace{-6pt}


\begin{columns}
\column{.5\textwidth}
\uncover<2->{\begin{block}{Wrong proof, wrong answer}
\fontsize{11pt}{17pt}\selectfont\normalfont
$R'(x)\overset{?}{=}2\pi \sum  \cos(2\pi n^2x)$ 
does not converge for any $x\in\R$.
For $x$ irrational, $\cos(2\pi n^2x)$ is dense in $[-1,1]$.  For $x\in\Q$ it oscillates.
\end{block}}
\column{.5\textwidth}
\uncover<3->{\begin{block}{Wrong proof, right answer}
$z = x+iy$

$R'(x)=\Big(\lim_{y\to 0^+}\Im \sum\frac{e(n^2z)}{n^2}\Big)'$

$R'(x)\overset{?}{=} \pi\lim_{y\to 0^+}\Re \big(\theta(z)-1\big)$

\vspace{4pt}

$\exists\lim_{y\to 0^+}\Re\theta(z)\Leftrightarrow x=\frac{a}{2b}$, {\small $2\nmid b$}.
\end{block}}
\end{columns}


\end{frame}




%%%%%%%%%%
% SLIDE %%
%%%%%%%%%%
\begin{frame}



\begin{tabular}{c}
\includegraphics[scale=0.241]{./images/r_1_2.png}
\\
{\small close to $\mathsf{x=1/2}$}
\end{tabular}
\begin{tabular}{c}
\includegraphics[scale=0.241]{./images/r_5_14.png}
\\
{\small close to $\mathsf{x=5/14}$}
\end{tabular}

\

\

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\small Some harmonic analysts dub as \emph{chirps} this kind of oscillations resembling to the function $x^\alpha\sin x^{-\beta}$.
\end{block}}



\end{frame}





%%%%%%%%%%
% SLIDE %%
%%%%%%%%%%
\begin{frame}

\[
 R(x)=\sum_{n=1}^\infty
\frac{\sin(2\pi n^2x)}{n^2}
\]

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\begin{eqnarray*}
\text{Poisson's summation}&&
\sum_{n\in\Z} e^{-\pi n^2t}
=\frac{1}{\sqrt{t}}\sum_{n\in\Z} e^{-\pi n^2/t}
\\
&\updownarrow&
\\
\text{Modular relation}&&
\theta(z)=\sqrt{\frac{i}{2z}}\;\theta\big(\frac{-1}{4z}\big)
\end{eqnarray*}
\end{block}}


{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\[
\text{Analytic tools}
\quad\rightarrow\quad
R(h)-R(0)= C h^{1/2}+O(h).
\]
\end{block}}


\end{frame}


%%%%%%%%%%
% SLIDE %%
%%%%%%%%%%
\begin{frame}



{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\begin{center}
\begin{tabular}{c}
\includegraphics[scale=0.481]{./images/r_g.png}
\\
\color{blue}Graph of the Riemann example
\end{tabular}
\end{center}
\end{block}}



\end{frame}



%%%%%%%%%%
% SLIDE %%
%%%%%%%%%%
\begin{frame}


{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\[
 R(x)=\sum_{n=1}^\infty
\frac{\sin(2\pi n^2x)}{n^2}
\]
\end{block}}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
Expansion at the cusps (Poisson's summation in arithmetic progressions), for $h>0$
\[
R\big(\frac aq+h\big)
-R\big(\frac aq\big)
=q^{-1}G h^{1/2}
+
O\big( h q^{1/2}\big)
\]
with $G$ essentially a Gauss sum.
\end{block}}


{\setbeamercolor{block title}{bg=white}
\begin{block}{}
Diophantine approximation $\rightarrow$ characterization of the H\"older exponent in terms of the continued fraction. 
\end{block}}


\end{frame}



%%%%%%%%%%
% SLIDE %%
%%%%%%%%%%
\begin{frame}





\begin{tabular}{c}
\includegraphics[scale=0.241]{./images/r_1_2.png}
\\[-37pt]
{\color{black}\small$\mathsf{x=1/2}$}
\\[17pt]
\
\end{tabular}
\begin{tabular}{c}
\includegraphics[scale=0.241]{./images/r_1_3.png}
\\[-37pt]
{\color{black}\small$\mathsf{x=1/3}$}
\\[17pt]
\
\end{tabular}

%\hrulefill

\begin{tabular}{c}
\includegraphics[scale=0.241]{./images/r_2_5.png}
\\[-37pt]
{\color{black}\small$\mathsf{x=2/5}$}
\\[7pt]
\
\end{tabular}
\begin{tabular}{c}
\includegraphics[scale=0.241]{./images/r_1_4.png}
\\[-37pt]
{\color{black}\small$\mathsf{x=1/4}$}
\\[7pt]
\
\end{tabular}



\end{frame}



%%%%%%%%%%
% SLIDE %%
%%%%%%%%%%
\begin{frame}

\[
R\big(\frac aq+h\big)
-R\big(\frac aq\big)
=q^{-1}G h^{1/2}
+
O\big( h q^{1/2}\big)
\]
{\setbeamercolor{block title}{bg=white}
\begin{block}{}
To give a chance to the differentiability the Gauss sum has to vanish. This happens when $q\equiv 2\pmod{4}$.
\end{block}}

\pause

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
In the rest of the cases, we have something like $q^{-1/2}h^{1/2}$. The typical approximation by rationals has $h\approx q^{-2}$. Then for the most of the points the expected  H\"older exponent is $3/4$ and $d(3/4)=1$.
\end{block}}

\pause 

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
Quicker approximations by rationals, say $h\approx q^{-r}$, give $h^{1/2+1/2r}$. If $r$ is large, this suggests $d(1/2)=0$ and a continuous variation with $r$.
\end{block}}



\end{frame}



%%%%%%%%%%
% SLIDE %%
%%%%%%%%%%
\begin{frame}


{\setbeamercolor{block title}{bg=white}
\begin{block}{}
Jaffard (1996) $\rightarrow$ The function $R(x)$ is multifractal.
\end{block}}


{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\[
 R_\alpha(x)=\sum_{n=1}^\infty
\frac{\sin(2\pi n^2x)}{n^\alpha},
\qquad\alpha>1.
\]
\end{block}}

\vspace{-6pt}

\begin{center}
\begin{tabular}{c}
\includegraphics[scale=0.3]{./images/jafspect.png}
\end{tabular}
\end{center}


{\setbeamercolor{block title}{bg=white}
\begin{block}{}
The general quadratic case is treated with other $\theta$-functions.
\end{block}}


\end{frame}


%%%%%%%%%%
% SLIDE %%
%%%%%%%%%%
\section{Automorphic}
\begin{frame}
\frametitle{The automorphic setting}



{\setbeamercolor{block title}{bg=white}
\begin{block}{}
J.-P. Serre, H.M. Stark (1977): $\theta$-functions and modular forms of weight $1/2$ are the same thing.

\

quadratic case $\leftrightarrow$ weight $1/2$
\end{block}}


{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\textbf{Natural question:}
What happen with other weights?
\end{block}}

\begin{block}{}
\[
f(z)=\sum a_ne(nz)\quad\text{automorphic}
\]

\[
\text{fractional integral\quad $\rightarrow$\quad}f_\alpha(x)=\sum \frac{a_n}{n^\alpha}e(nx).
\]
\end{block}


\end{frame}



%%%%%%%%%%
% SLIDE %%
%%%%%%%%%%
\begin{frame}
\frametitle{Some results}


\begin{block}{}
\underbar{Ch. (2004)} If $f$ is a cusp form, $f_\alpha(x)$ is pure fractal and not multifractal.
\end{block}

\pause

\begin{block}{}
\underbar{Petrykiewicz (2013)} If $f$ is a classical Eisenstein series then the H\"older exponent can be determined at every point in some ranges.
\end{block}

\pause

\begin{block}{}
\underbar{Ch. {\&} Ruiz-Cabello (2014)}  If $f$ is not a cusp form, $f_\alpha$ is a multifractal under strict conditions on $\alpha$ and the weight.
\end{block}


\end{frame}


%%%%%%%%%%
% SLIDE %%
%%%%%%%%%%
\begin{frame}

{\small In the last result,  we only treat the case with $\beta_f\le 1$ and it forces the restrictions on the ranges. We plan to extend the result in a near future.}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\begin{center}
\begin{tabular}{c}
\includegraphics[scale=0.4]{./images/modspect.png}
\\
\color{blue}The spectrum of singularities of $f_\alpha$
\end{tabular}
\end{center}
{\footnotesize $w=$\;weight of the modular form.}
\end{block}}




\end{frame}


%%%%%%%%%%
% SLIDE %%
%%%%%%%%%%
\begin{frame}
\frametitle{Why the case of higher degree is completely different?}



{\setbeamercolor{block title}{bg=white}
\begin{block}{}
In the case of degree $k$, Poisson summation allows to control the (asymptotics of the) oscillation of the function  on each interval
\[
\big(\frac aq-h,\frac aq+h\big),
\qquad
h<\frac{1}{q^k}.
\]
If $k=2$, they define a covering of $\R$. In general, 
the union of these intervals has Hausdorff dimension $2/k$.
Very thin for $k>2$.
\end{block}}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
A more important barrier to treat the case $k > 2$ is that the error term after using  Poisson's summation, is better than trivial only when $h < q^{-k/2}$  while for the Diophantine approximation process we have to deal with $h$ almost like $q^{-1}$.
\end{block}}




\end{frame}


%%%%%%%%%%
% SLIDE %%
%%%%%%%%%%
\section{Theorem}
\begin{frame}
\frametitle{What have we actually proved?}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\[
 F(x)=\sum_{n=1}^\infty\frac{e\big(P(n)x\big)}{n^{\alpha}}.
\]
\end{block}}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
Define $\nu_0=\max(\nu_F,2)$ with $\nu_F$ the maximal multiplicity of a zero of $P'$.
\end{block}}



\begin{block}{Theorem}
For $1+k/2<\alpha<k$, $0\le \beta<1/2k$,
\[
d\big(\beta+\frac{\alpha-1}{k}\big)
\ge (\nu_0+2)\beta.
\]
\end{block}


\end{frame}



%%%%%%%%%%
% SLIDE %%
%%%%%%%%%%
\begin{frame}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\[
 F(x)=\sum_{n=1}^\infty\frac{e\big(P(n)x\big)}{n^{\alpha}}.
\]
\end{block}}


\begin{block}{}
We have also an upper bound. We think that in the \lq\lq typical case'' the lower bound gives the true dimension.
\end{block}

\vspace{-10pt}
\begin{center}
\begin{tabular}{c}
\\[2pt]
\includegraphics[scale=0.231]{./images/mainspect.png}
\\
\color{blue}Proved
\\
$F$ is multifractal
\end{tabular}
\hspace{-10pt}
\begin{tabular}{c}
\includegraphics[scale=0.286]{./images/conjspect.png}
\\
\color{blue}Conjectured in the ``typical case''
\end{tabular}
\end{center}

\end{frame}






%%%%%%%%%%
% SLIDE %%
%%%%%%%%%%
\section{Heuristics}
\begin{frame}
\frametitle{What are the ideas under the proof?}


{\setbeamercolor{block title}{bg=white}
\begin{block}{}
The proof of the theorem is long and technical and involves several different ingredients (Poisson summation, variants of large sieve, Tur\'an's method, Weyl's inequality, generalized Cantor sets\dots)
\end{block}}

\

{\color{pink}Instead of reviewing the steps and the role of these ingredients in the proof we try to motivate the result.}

\ 

\pause

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
The subsequent slides do \underbar{not} reflect the scheme of the actual proof but they give some hints about the underlying ideas. The point is
\begin{center}
\color{blue} How could one imagine the result?
\end{center}
\end{block}}

\end{frame}

%%%%%%%%%%
% SLIDE %%
%%%%%%%%%%
\begin{frame}



{\setbeamercolor{block title}{bg=white}
Assume $h>0$ and write $k=\deg P$
\begin{block}{}
\[
F\big(\frac aq+h\big)
-
F\big(\frac aq\big)
=
\sum 
e\big(\frac{aP(n)}q\big)
w_h(n),
\quad
w_h(n)=\frac{e\big(hP(n)\big)-1}{n^\alpha}
\]
\end{block}}

$w_h$ oscillates for $n>h^{-1/k}$, and it is small for $n$ small. The bulk of the contribution comes from $n\asymp h^{-1/k}$ where $w_h(n)\asymp h^{\alpha/k}$.


\begin{block}{Model}
If $h= q^{-r}$, $r>k$, $F(a/q+h)-F(a/q)$ should behave like
\[
h^{\alpha/k}\sum_{n\asymp q^{r/k}} 
e\big(\frac{aP(n)}q\big).
\]
\end{block}




\end{frame}




%%%%%%%%%%
% SLIDE %%
%%%%%%%%%%
\begin{frame}



{\setbeamercolor{block title}{bg=white}
\begin{block}{}
The analytic theory of exponential sums appeals very often to the heuristics of \emph{square root cancellation}.
\end{block}}

{\color{gray}[The (unproved, out of reach) $(\epsilon,1/2+\epsilon)$ exponent pair]}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
It would solve the Gauss circle problem, the Lindel\"of hypothesis for~$\zeta$, etc.
\end{block}}


\begin{block}{Rough idea in many contexts}
A really oscillatory exponential sum $S$ with $N$ terms should verify $|S|\le C_\epsilon N^{1/2+\epsilon}$, and typically $|S|\ge C_\epsilon' N^{1/2-\epsilon}$.
\end{block}

\[
 \text{Quadratic Gauss sum}
 \quad
 \sum_{n=1}^qe\big(\frac{n^2}{q}\big)
 =
 \frac{1+(-i)^q}{1-i}\sqrt{q}.
\]



\end{frame}




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% SLIDE %%
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\begin{frame}



{\setbeamercolor{block title}{bg=green}
\begin{block}{{\color{black}Model}}
If $h= q^{-r}$, $r>k$, $F(a/q+h)-F(a/q)$ should behave like
\[
h^{\alpha/k}
\sum_{n\asymp q^{r/k}} 
e\big(\frac{aP(n)}q\big).
\]
\end{block}}


{\setbeamercolor{block title}{bg=white}
\begin{block}{}
Note that $e\big(aP(n)/q\big)$ is $q$-periodic. The square root cancellation heuristics applied to each $q$-block, leads to
\[
h^{\alpha/k}q^{r/k-1} q^{1/2}= h^{(\alpha-1)/k+1/2r}
\]
\end{block}}


{\setbeamercolor{block title}{bg=pink}
\begin{block}{{\color{blue}Jarn\'{\i}k-Besicovitch theorem}}
\[
\dim\Big\{x\;:\; \big|x-\frac aq\big|<q^{-r}\text{ infinitely often}\Big\}=\frac{2}{r}
\]
\end{block}}





\end{frame}




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% SLIDE %%
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\begin{frame}



{\setbeamercolor{block title}{bg=green}
\begin{block}{}
If $h= q^{-r}$, $r>k$, $F(a/q+h)-F(a/q)$ should behave like
\[
h^{\alpha/k}\sum_{n\asymp q^{r/k}} 
e\big(\frac{aP(n)}q\big)
\approx
h^{(\alpha-1)/k+1/2r}.
\]
\end{block}}


{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\[
 \text{H\"older exponent }
\frac{\alpha-1}{k}+\frac{1}{2r}
 \text{ in a set of }
\dim=\frac{2}{r}
\]
\end{block}}

\pause

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
Writing $\beta =1/2r$ we have the lower bound
\[
d\big(\beta+\frac{\alpha-1}{k}\big)
\ge 4\beta
\]
that is part of the theorem.
\end{block}}



\end{frame}




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% SLIDE %%
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\begin{frame}


{\setbeamercolor{block title}{bg=white}
\begin{block}{}
This bound can be improved if $P$ has  multiple zeros because in this case the square root cancellation is violated. 
\end{block}}


{\setbeamercolor{block title}{bg=white}
\begin{block}{{\color{black}{\bf Model example:}\quad\  $q=p^k$, $p\nmid a$,  $p\nmid k$}}
\[
\sum_{n=1}^q 
e\big(\frac{a n^k}q\big)
= q^{1-1/k}
\]
\end{block}}


{\setbeamercolor{block title}{bg=white}
\begin{block}{}
In this case
\[
h^{\alpha/k}\sum_{n\asymp q^{r/k}} 
e\big(\frac{aP(n)}q\big)
\approx
h^{(\alpha-1)/k+1/kr}
\]
\end{block}}




\end{frame}





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% SLIDE %%
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\begin{frame}


{\setbeamercolor{block title}{bg=white}
\begin{block}{}
In general, if $\nu >1$ the maximal multiplicity of a zero of $P'$, and $q=p^{\nu+1}$, the model suggests that  
if $h= q^{-r}$, $r>k$, $F(a/q+h)-F(a/q)$ should behave like
\[
h^{\alpha/k}\sum_{n\asymp q^{r/k}} 
e\big(\frac{aP(n)}q\big)
\gg 
h^{(\alpha-1)/k+1/(\nu+1)r}.
\]
\end{block}}



{\setbeamercolor{block title}{bg=white}
\begin{block}{{\color{black}(Modified) Jarn\'{\i}k-Besicovitch theorem}}
\[
\dim\Big\{x\;:\; \big|x-\frac aq\big|<q^{-r}, \ q=p^{\nu+1}\text{ inf. often}\Big\}
=
\frac{1}{r}
+\frac{1}{(\nu+1)r}.
\]
\end{block}}


\end{frame}



%%%%%%%%%%
% SLIDE %%
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\begin{frame}



{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\[
 \text{H\"older exponent }
\frac{\alpha-1}{k}+\frac{1}{(\nu+1)r}
 \text{ in a set of }
\dim=
\frac{1}{r}
+\frac{1}{(\nu+1)r}
\]
\end{block}}


\pause 

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
Writing $\beta =1/(\nu+1)r$ we have the lower bound
\[
d\big(\beta+\frac{\alpha-1}{k}\big)
\ge (\nu+2)\beta
\]
that is the remaining part of the theorem.
\end{block}}


\end{frame}





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% SLIDE %%
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\section{Proof}
\begin{frame}
\frametitle{What are the main steps in the actual proof?}

{\setbeamercolor{block title}{bg=white}
\begin{block}{\color{black}\bf 1. Local analysis}
\underbar{\rm Tools}:
Poisson's summation, Weyl's inequality.

\underbar{\rm Output}:
Approximation around rational values.
\end{block}}

\pause

{\setbeamercolor{block title}{bg=white}
\begin{block}{\color{black}\bf 2. Average oscillation}
\underbar{\rm Tools}: Variants of large sieve inequality.

\underbar{\rm Output}:
Some control of the oscillation averaging over thin sets.
\end{block}}

\pause

{\setbeamercolor{block title}{bg=white}
\begin{block}{\color{black}\bf 3. Lower bounds for exponential sums}
\underbar{\rm Tools}: Tur\'an's method, polynomials over finite fields.

\underbar{\rm Output}: 
General lower bound for families of exponential sums.
\end{block}}




\end{frame}


%%%%%%%%%%
% SLIDE %%
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\begin{frame}
\frametitle{What are the main steps in the actual proof?}


{\setbeamercolor{block title}{bg=white}
\begin{block}{\color{black}\bf 4. Some Cantor-like sets}
\underbar{\rm Tools}: Elementary measure theory.

\underbar{\rm Output}:
Dimension of general limit set of sequences of nested intervals.
\end{block}}

\pause

{\setbeamercolor{block title}{bg=white}
\begin{block}{\color{black}\bf 5. Construction of the fractal set}
\underbar{\rm Tools}: Jarn\'{\i}k-Besicovitch, Diophantine approximation.

\underbar{\rm Output}:
A fractal set in which the H\"older exponent is constant.
\end{block}}




\end{frame}

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\section{Expectations}
\begin{frame}
\frametitle{What is the theorem that we have not proved?}


{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\begin{itemize}
 \item 
Some artificial examples with unbalanced multiplicities produce a serious obstruction to the square root cancellation philosophy and its modification. We have not a clear conjecture for the general case but it seems to be exact for most polynomials (including those with successive derivatives having simple zeros). 
\end{itemize}
\end{block}}

\pause

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\begin{itemize}
 \item 
Our approach only can deal with irrationals very well approximated by rationals. We expect maximal cancellation in the rest of the cases.
\end{itemize}
\end{block}}

\pause

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\begin{itemize}
 \item 
The behavior in the rationals depends on the non-vanishing of 
a finite exponential sum.
\end{itemize}
\end{block}}





\end{frame}



%%%%%%%%%%
% SLIDE %%
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\begin{frame}


{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\begin{itemize}
 \item 
 Near to $\Q$: Cancellation in terms of multiplicities.
 \item
 Far away from $\Q$: Maximal cancellation.
 \item
 In $\Q$: Arithmetical properties.
\end{itemize}
\end{block}}




{\setbeamercolor{block title}{bg=green}
\begin{block}{}
\begin{center}
\begin{tabular}{c}
\includegraphics[scale=0.36]{./images/conjspect.png}
\\
\color{blue}Conjecture for the ``typical case''
\end{tabular}
\end{center}
\end{block}}



\end{frame}







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\section{Bibliography}
\begin{frame}
\frametitle{Some related papers}

\begin{itemize}%\setlength{\itemsep}{-3pt}

\item
F.~Chamizo and S.~Ruiz-Cabello.
Modular forms and multifractal {F}ourier series.
In preparation.

\item
F.~Chamizo and A.~Ubis.
Multifractal behavior of polynomial {F}ourier series.
{\em Adv. Math.}, 250:1--34, 2014.


\item
F.~Chamizo and A.~Ubis.
Some {F}ourier series with gaps.
{\em J. Anal. Math.}, 101:179--197, 2007.

\item
F.~Chamizo.
Automorphic forms and differentiability properties.
{\em Trans. Amer. Math. Soc.}, 356(5):1909--1935, 2004.

\item
F.~Chamizo and A.~C{\'o}rdoba.
Differentiability and dimension of some fractal {F}ourier series.
{\em Adv. Math.}, 142(2):335--354, 1999.

\end{itemize}



\end{frame}


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\begin{frame}

\begin{center}
{\LARGE Presentation available in}

\

{\LARGE\url{http://www.uam.es/fernando.chamizo}}
\end{center}

\end{frame}



\end{document}
