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\title[Fourier series with gaps and arithmetic]{Fourier series with gaps and arithmetic}
\institute{S\'eminaire Cristolien d'Analyse Multifractale\\ Universit\'e Paris-Est Cr\'eteil}
\date{21 mars 2019}
\author[F. Chamizo]{Fernando Chamizo (UAM-ICMAT)}


\begin{document}

\begin{frame}[noframenumbering]
\titlepage
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\section{\texorpdfstring{Riemann's example}{Riemann's example}}
\begin{frame}%\pcount
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To be or not to be
{\dots}
differentiable
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\begin{miblock}
According to Weierstrass, Riemann claimed that $f$ is nowhere differentiable
\end{miblock}
\vspace{-15pt}

\[
 f(x)=
 \sum_{n=1}^\infty\frac{\sin(\pi n^2 x)}{n^2}
 \ 
 \begin{tabular}{c}
 \includegraphics[height=143.149pt, keepaspectratio=true]{./images/rie_examp.png}
 \end{tabular}
\]
\vspace{-10pt}


\begin{blthree}
Quick conclusion: Riemann and Weierstrass did not have a computer.
\end{blthree}


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\frametitle{Two fake proofs}
\vfill




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    \node[rounded corners,fill=red!10!white,draw=masrojo,line width=.5pt] at (2.2,1.1)
        {Wrong proof, wrong answer};
    \node[rounded corners,fill=white,draw=ct2,line width=.5pt] at (2.2,-1.4)
        {\begin{tabular}{p{125pt}}
         $f'(x)=\pi\sum \cos(\pi n^2 x)$\phantom{mm}
         does not converge for any real number $x$
         \quad$\overset{(?)}{\Rightarrow}$\quad
         $f$ is nowhere differentiable 
         and Riemann's alleged claim is right.
        \end{tabular}
        };
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        {Wrong proof, right answer};
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        {\begin{tabular}{p{125pt}}
        $f'(x)$ is the derivative of
        $\lim_{y\to 0}\sum e^{-\pi n^2y}\frac{\sin(\pi n^2 x)}{n^2}$
        $=\pi\lim_{y\to 0}\frac{1}{2}\big(\vartheta(x+iy)-1\big)$
        and it exists and takes the value $-\pi/2$ exactly at the \emph{\color{black!90!white}cusps} 
        $x=a/b$, $2\nmid a,b$.
        \end{tabular}
        };
\end{tikzpicture}

\

$\vartheta(z)=\sum_{n\in\Z}e(\frac 12 n^2z)$ (Jacobi)
\hfill {\bf Notation:}
$\boxed{e(x)=e^{2\pi i x}}$


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\begin{frame}
\frametitle{A couple of unexpected connections}
\vfill


\begin{miblock}
\includegraphics[height=8.478pt, keepaspectratio=true]{./images/1.png}\  Riemann's example appears in the asymptotics of the sum of the Jacobi symbol
$\displaystyle{\sum_{2\nmid n<X}\sum_{2\nmid m<Y}
\Big(\frac{n}{m}\Big)}$.
\end{miblock}
\vspace{-5pt}

{\tiny Conrey, J. B.; Farmer, D. W.; Soundararajan, K.
Transition mean values of real characters.
J. Number Theory 82 (2000), no. 1, 109--120.
}

\medskip

\begin{miblock}
\includegraphics[height=8.478pt, keepaspectratio=true]{./images/2.png}\  Riemann's example appears in the evolution of a polygonal vortex under a fluid dynamic model.
\end{miblock}
\vspace{-5pt}

{\tiny de la Hoz, F.; Vega, L. Vortex filament equation for a regular polygon. Nonlinearity 27 (2014), no. 12, 3031--3057. 
}

\


\begin{blthree}
{\footnotesize Square frequencies are linked to Schr\"odinger equation and it opens the opportunity for several applications, for instance the quantum Talbot effect.}
{\tiny Berry, M. V.; Klein, S.
Integer, fractional and fractal Talbot effects.
J. Modern Opt. 43 (1996), no. 10, 2139--2164}, 
{\tiny (Vega, L.; Eceizabbarrena, D. Work in progress)}
\end{blthree}



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\section{\texorpdfstring{Gaps}{Gaps}}
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Gaps: Good or bad?
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\begin{frame}
\frametitle{A couple of unexpected connections}
\vfill


\begin{miblock}
Riemann's example is only differentiable in a meager set and Weierstrass constructed his fractal-like functions with lacunary Fourier series. Do gaps imply chaos?
\end{miblock}

\begin{blthree}
Some theorems suggests that lacunary frequencies have a smoothing effect. 
\par 

Namely for $f\sim \sum e(n_jx)$ with $n_{j+1}/n_j>c>1$ it is known:
\end{blthree}


\begin{miblock}
 If $f$ is bounded then its Fourier series is absolutely convergent.
\end{miblock}


\begin{miblock}
 If $f$ is integrable then it belongs to $L^2$. In particular $\sum |c_n|^2<\infty$.
\end{miblock}

\begin{blthree}
These results are blatantly false if lacunarity is dropped.
\end{blthree}


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\begin{block}{}
{\color{masrojo}\fbox{Conjecture}} {\footnotesize\color{blue}(Hardy, Littlewood, Rudin, C\'ordoba)}:
The ``kernel'' $\sum e(n^2 x)$ is a multiplier $L^2\longrightarrow L^p$ for $p<4$ i.e., $L^2$ Fourier series with squares frequencies are automatically in $L^p$ for $p<4$.
\end{block}

\begin{block}{}
It implies Rudin's conjecture: An arithmetic progression of length $N$ may contain at most $O\big(N^{1/2+\epsilon}\big)$ squares. 
\end{block}

\


\begin{blthree}[colback=yellow!20!white, colframe=yellow!80!white]
{\bf Uncertainty principle:} To examine details at level $\epsilon$ we need a frequency range of at least $\epsilon^{-1}$.
\hfill  $[x,p]=i\hbar$,\quad $4\pi \|xf\| \|\xi\hat{f}\|\ge \|f\|^2$
\end{blthree}


\medskip 

\begin{bltwo}
The gaps give us an opportunity in our everlasting fight against uncertainty principle. On the other hand the possible resonances are sometimes related to deep arithmetical problems.
\end{bltwo}

\medskip 


\begin{blthree}
Intricate theory of exponential sums estimation in number theory.
\end{blthree}


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\section{\texorpdfstring{Multifractal}{Multifractal}}
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How chaotic is your function?
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\vfill
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\vfill


\begin{miblock}
 {\bf Theorem} (Jaffard, 1996).
 Riemann's example and its fractional  derivatives are multifractals.
\end{miblock}

\begin{miblock}
 \[
\includegraphics[height=59.113pt, keepaspectratio=true]{./images/jafspect.pdf}
 \]
\end{miblock}

\

\begin{bltwo}
 {\bf Theorem} (Ch. Ubis, 2014).
 The functions $\sum n^{-\alpha} e\big(P(n)\big)$ with $P\in\Z[x]$, $\deg P>2$ are  also multifractal in some ranges.
\end{bltwo}

\begin{blthree}
 The higher gaps for $\deg P>2$ do not payback because one loses self-similarity.
\end{blthree}



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\section{\texorpdfstring{A BMO series}{A BMO series}}
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Visiting a Riemann's relative
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\vfill
\begin{block}{}
 We consider a relative of Riemann's example at the edge of convergence
 \[
  F(x)
  =
  \sum_{n=1}^\infty
  \frac{e(n^2x)}{n}.
 \]
\end{block}

\begin{blthree}
{\bf Remarks}
\medskip

 {\large
 \includegraphics[height=9.113pt, keepaspectratio=true]{./images/1.png}\  It clearly diverges in a dense set of rationals.
 \medskip 
 
 \includegraphics[height=9.113pt, keepaspectratio=true]{./images/2.png}\  It nevertheless defines a BMO function.
 }
\end{blthree}

\begin{bltwo}
{\small\bf Our basic naive question}
\medskip

{\large How much does it differ from being a bounded function?}
\end{bltwo}



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BMO, an interlude

{\large(when mean oscillation is not mean)}
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\begin{miblock}[colback=yellow!20!white, colframe=yellow!80!white]
 The space of \textbf{B}ounded \textbf{M}ean \textbf{O}scillation was introduced by John and Nirenberg in 1961 and readily applied by Moser to get the regularity of some elliptic PDEs. 
 Its relevance was boosted by C. Fefferman duality theorem $(H^1)^* =\text{BMO}$. 
\end{miblock}

\medskip

\begin{miblock}
 \[
 f\in \text{BMO}
 \quad\text{means}\quad
  \|f\|_I
  =
  \frac{1}{|I|}
  \int_I|f-f_I|<C
 \]
 uniformly on intervals $I$, where $f_I$ is the average of $f$. 
\end{miblock}

\medskip

\begin{blthree}
 {\bf Comment.}
 $f:[a,b]\longrightarrow\R$ bounded $\Rightarrow$ $f\in\text{BMO}$.
\end{blthree}


\begin{blthree}
 {\bf (Counter)example.}
 $\log x$ as a function $(0,1]\longrightarrow\R$ belongs to \text{BMO}. 
\end{blthree}


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\vfill

\begin{itemize}
 \item
 From the point of view of harmonic analysis is a substitute for $L^\infty$ well-behaved under interpolation and duality. 
 \item
 By John-Nirenberg inequality, roughly speaking, $\text{BMO}$ is like $L^\infty$ allowing at most logarithmic singularities.
\end{itemize}

\medskip

\begin{blthree}
 {\bf Theorem.}
 (Ch., C\'ordoba, Ubis 2019) 
 If $\nu_{n+1}/\nu_n\ge1+\delta\max(|a_n|, |a_{n+1}|)$
 \[
  \limsup_{|I|\to 0}
  \Big\|
  \sum_{n=1}^\infty a_ne(\nu_n x)
  \Big\|_I
  \le \frac{3}{\delta}\root{3}\of{12\pi}.
 \]
\end{blthree}

{\small
\[
  \sum_{n=1}^\infty
  \frac{e(n^kx)}{n}
  \in\text{BMO}
  \quad
  \text{with some control on its ``\text{BMO} norm''}.
\]
}

{\small
Paradox: Bigger gaps $\to$ closer to vanishing mean oscillation.
}



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\section{\texorpdfstring{Results}{Results}}
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Converging to something
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\begin{frame}
\frametitle{Our results (joint with  C\'ordoba and Ubis 2019)}
\vfill


\vspace{-20pt}

\begin{block}{}
 \[
  F(x)
  =
  \sum_{n=1}^\infty
  \frac{e(n^2x)}{n}
 \]
\end{block}


\


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    \node[rounded corners,fill=white,draw=ct2,line width=.5pt] at (10pt,-10pt)
        {\bf \color{blue}Rough formulation};
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%{\bf \setlength{\fboxrule}{1.5pt}\setlength{\fboxsep}{1pt}\fbox{\colorbox{green!10!white}{Rough formulation}}}

\smallskip

\begin{bltwo}
 {\bf Theorem 1.}
 Full characterization of the convergence set of the series.
\end{bltwo}


\begin{blthree}
 Seuret, Ubis 2017 (Ann. Inst. Fourier 67, no. 5, 2237--2264) proved the convergence and the divergence for classes of irrational values.  
\end{blthree}

\medskip

\begin{bltwo}
 {\bf Theorem 2.}
 Precise estimation of the oscillation in small intervals.
\end{bltwo}


\begin{blthree}
  John, Nirenberg 1961 (Comm. Pure Appl. Math. 14 415--426)  proved that 
  $\frac{1}{|I|}\big|\{x\in I:|f(x)-f_I|>\lambda\}\big|$
  is $O\big(e^{-C\lambda}\big)$ for $\text{BMO}$ functions. 
\end{blthree}



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\frametitle{Our results (joint with  C\'ordoba and Ubis 2019)}
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\smallskip

\begin{bltwo}
 {\bf Theorem 1.}
 For $x$ irrational, the series converges if and only if 
 \vspace{-5pt}
 \[
  \frac{1}{2}\sum_{j=1}^\infty
  \frac{\theta_{p_j/q_j}}{\sqrt{q_j}}\log \frac{q_{j+1}}{q_j}\quad \text{does}. 
 \]
 \vspace{-10pt}
 
 In fact the difference between $F$ and this sum is bounded in the convergence set.
 Here  $p_j/q_j$ are the convergents of the continued fraction of $x$ and $\theta_{p_j/q_j}$ are the normalized Gauss sums.

\end{bltwo}


\medskip

{\small
\begin{itemize}
 \item 
 $\theta_{p_j/q_j}\in\{0,\pm 1, \pm i, \pm 1\pm i\}$.
 Then $\frac{\theta_{p_j/q_j}}{\sqrt{q_j}}$ has an exponential decay. 
 \item
 Corollary (Seuret, Ubis). If $q_{j+1}\ne o\big( q_j \exp\big(q_j^{1/2}\big)\big)$ for $2\nparallel q_j$, it diverges.
\end{itemize}
}


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\frametitle{Our results (joint with  C\'ordoba and Ubis 2019)}
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\medskip

\begin{bltwo}
 {\bf Theorem 2.}
 There are constants $c_1, c_2, C>0$ such that
 \[
  C^{-1}e^{-c_2\lambda\sqrt{q}}
  \le 
  \frac{1}{|I|}
  \big|\{x\in I:|F(x)-F_I|>\lambda\}\big|
  \le 
  Ce^{-c_1\lambda\sqrt{q}}
 \]
 for $I$ the interval of real numbers with continued fraction extending that of $p/q$. In fact $c_1=c_2$ if $4\mid q$.
\end{bltwo}


{\small
\begin{itemize}
 \item 
 $|I|$ is comparable to and interval centered at $p/q$ of length $q^{-2}$. 
 \item
 The upper bound still holds for any subinterval of $I$.
 \item 
 The oscillation depends on the Diophantine approximation, being exponentially small on the denominator in the Farey dissection. 
\end{itemize}
}




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\section{\texorpdfstring{Poisson}{Poisson}}
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A modular forms as \mbox{Poisson} summation in disguise 
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\emph{Poisson summation for number theory is what a car is for people in modern communities --it transports things to other places and it takes you back home when applied next time-- one cannot live without it.}
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\begin{tikzpicture}
\invisible{\draw[step=1cm,help lines] (-.35,.7) grid (10.35,-.5);}
\node[anchor=south east,align=right] at (10.35,0) {H. Iwaniec, E. Kowalski};
\node[anchor=north east,align=right,colortema2!70!black] at (10.35,0) {\emph{Analytic Number Theory}};
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{\small \color{RoyalPurple}Some avatars of the invariance  of Gaussians under the Fourier transform:}\

\begin{miblock}[colback=SkyBlue!15!white, colframe=SkyBlue!80!white]
\[
 \sum_{n\in\Z} e^{-\pi tn^2}
 =
 \frac{1}{\sqrt{t}}
 \sum_{n\in\Z} e^{-\pi n^2/t}
\]
\hrulefill
\[
 \theta(z)
 \text{ is a half weight modular form under the theta group}
\]
\hrulefill
\[
 \sideset{}{'}\sum_{n\le N}
 e\big(\frac{1}{2}n^2 x\big)
 =
 \frac{e(1/4)}{\sqrt{x}}
 \sideset{}{'}\sum_{n\le Nx}
 e\big(-\frac{n^2}{2x}\big)
 +O\Big(\frac{1}{\sqrt{x}}\Big)
\]
\end{miblock}

\begin{blthree}
 The last one establishes the connection with continued fractions. 
 
 \vspace{5pt}

 {\tiny Fiedler,  Jurkat,  K\"orner. Acta Arith. 32 (1977),  129--146.
 
 Hardy, Littlewood. Acta Math. 37:155--191, 193--239, 1914.}
\end{blthree}


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\begin{miblock}
Poisson summation in arithmetic progressions proves 
\[
 \sum_{q\le n<Q} e(n^2x)
 =
 \frac{\theta_{p/q}}{\sqrt{q}}
 \int_q^Q e(ht^2)\; dt + O(q^{1/2})
\]
with $p/q$, $P/Q$ are consecutive convergents of $x$ and $h=x-p/q$. 
\end{miblock}


\begin{miblock}
Replacing $Q$ by $N<Q$, applying partial summation 
\[
 \sum_{q\le n<Q} \frac{e(n^2x)}{n^2}
 =
 \frac{\theta_{p/q}}{\sqrt{q}}
 \int_q^Q \frac{e(ht^2)}t\; dt + O(q^{-1/2}).
\]
\end{miblock}



\begin{block}{Outcome:}
Characterization of the convergence.  
\end{block}

\begin{blthree}
 Just take $q=q_j$, $Q=q_{j+1}$, estimate the integral and sum in $j$.
\end{blthree}



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\begin{miblock}
 The average $F_I$ gives a blur version of $F$ at scale 
 $q^{-2}=q_{j_0}^{-2}$, the size of $I$.   
\end{miblock}

\begin{miblock}
 The uncertainty principle suggests that $\sum_{n^2< q^2} e(n^2x)/n^2$ contains alike information.  
\end{miblock}

\begin{miblock}
 If this is so, the oscillation is represented by $\sum_{n>q} e(n^2x)/n^2$.
\end{miblock}



\begin{block}{Outcome:}
\[
 F(x)-F_I 
 =
 \frac 12\sum_{j\ge j_0} \frac{\theta_{p_j/q_j}}{\sqrt{q_j}} \log \frac{q_{j+1}}{q_j}
 +
 O\Big(\frac{1}{\sqrt{q_{j_0}}}\Big)
 \qquad
 I = I_{p_{j_0}/q_{j_0}}.
\]
\end{block}



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\section{\texorpdfstring{Continued fractions}{Continued fractions}}
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Continued and discontinued fractions
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\begin{frame}
\frametitle{Two metric results after Khinchin}
\vfill

\begin{blthree}
 Although the partial quotients in the continued fraction are not exactly independent, they behave like that.
\end{blthree}


{\small $I=I_{p/q}$, $\frac pq=\frac{p_{j_0}}{q_{j_0}}$}


\begin{blthree}
 So we can control the measure when we specify a partial quotient{\dots}
\end{blthree}
\begin{bltwo}
 {\bf Lemma.}
 The measure of the $x\in I$ with a partial quotient $a_j=k$ for some fixed $j>j_0$ is comparable to  $k^{-2}|I|$
\end{bltwo}


\begin{blthree}
 Or even a tail sequence{\dots}
\end{blthree}



\begin{bltwo}
 {\bf Lemma.}
 
 If $A_n\ge 1$ and $S=\sum A_n^{-1}<\infty$ then 
 \vspace{-10pt}
 \[
  e^{c_1S}
  \le
  \frac{1}{|I|}
  \big|
  \{x\in I\;:\; a_{j_0+n}(x)\le A_n\text{ for }n\in\Z^+\}
  \big|
  \le e^{c_2S}.
 \]
\end{bltwo}




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\section{\texorpdfstring{Oscillation}{Oscillation}}
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Measuring vanishing sets
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        {\bf \color{blue}Recall};
\end{tikzpicture}

\begin{miblock}
\[
 F(x)-F_I 
 =
 \frac 12\sum_{j\ge j_0} \frac{\theta_{p_j/q_j}}{\sqrt{q_j}} \log \frac{q_{j+1}}{q_j}
 +
 O\Big(\frac{1}{\sqrt q}\Big)
 \qquad
 I = I_{p_{j_0}/q_{j_0}}.
\]
\end{miblock}


\


\begin{miblock}[colback=yellow!20!white, colframe=yellow!80!white]
We have $q_{j+1}/q_j=a_{j+1}+O(1)$ and according to the metric theory, the most of the $x\in I$ have not outrageously large partial quotients $\{a_{j_0+n}\}_{n\ge 2}$. 
\end{miblock}


\begin{blthree}
Note that $1/\sqrt{q_j}$ shows, at least, a geometric decay
\end{blthree}



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\begin{miblock}
Commonly 
\[
 F(x)-F_I 
 \approx
 \frac 12\frac{\theta_{p_{j_0}/q_{j_0}}}{\sqrt{q}} \log a_{j_0+1}
\]
and the approximation improves when $a_{j_0+1}$ grows because it becomes more dominant. 
\end{miblock}

\begin{block}{Outcome:}
 Large values of $|F(x)-F_I|$ require exponentially large partial quotients and this happens only in an exponentially small proportion of the interval. 
\end{block}



\begin{blthree}
Technical point: If $\theta_{p_{j_0}/q_{j_0}}$ vanishes, the expected dominant term is shifted. 
\end{blthree}


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\bf
Je vous remercie de votre attention!
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