View source for Littlewood problem

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The Littlewood problem for compatible Diophantine approximations is the question of the existence, for any real numbers $\alpha,\,\beta,\,\epsilon>0$, of a natural number $n$ such that $n \cdot \Vert n\alpha \Vert \cdot \Vert n\beta \Vert < \epsilon$, where $\Vert x \Vert$ is the distance from $x$ to the nearest integer. In certain cases, for example for rational $\alpha$ and $\beta$, and for numbers $\alpha$ and $\beta$ one of which can be represented by a [[Continued fraction|continued fraction]] with non-negative elements, the Littlewood problem has an affirmative answer.

The Littlewood problem for integrals is the problem whether for an arbitrary increasing sequence $M$ of natural numbers $(m_k)$, $k=1,2,\ldots$, one has
\begin{equation}\label{eq:1}
\int_0^1 \left\vert { \sum_{k=1}^n \exp(2\pi i m_k x) }\right\vert dx > f(n)
\end{equation}
with $f(n) = C \log n$, where $C > 0$ is an absolute constant and $n > n_0$. The following estimates have been obtained: either weaker estimates in comparison with \eqref{eq:1} for arbitrary sequences $M$, or estimates close to \eqref{eq:1} or even coinciding with this estimate, but for special sequences $M$.

The Littlewood problems were stated by J.E. Littlewood (see [[#References|[1]]]).

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  G.H. Hardy,   J.E. Littlewood,   "A new proof of a theorem on rearrangements"  ''J. London Math. Soc.'' , '''23'''  (1948)  pp. 163–168</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  J.W.S. Cassels,   "An introduction to the geometry of numbers" , Springer  (1959)</TD></TR>
</table>


====Comments====
The Littlewood problem for integrals has attracted the active interest of many mathematicians over a thirty year span. It was finally settled in the affirmative in 1981 by O.C. McGehee, L. Pigno and B. Smith [[#References|[a1]]], and, independently, by S.V. Konyagin [[#References|[a2]]]. A description of the problem just prior to its solution is given in [[#References|[a3]]], Sect. 1.3.

For $N=2n+1$ and $m_k = k$ the left-hand side of \eqref{eq:1} is equal to the [[Lebesgue constants|Lebesgue constant]] $L_N$.  As $L_N = (4/\pi^2)\log N + \lambda_N$, where the $\lambda_N$ are bounded and positive, it follows that the constant $C$ cannot be taken larger than $\frac{4}{\pi^2}$. A remaining conjecture is that \eqref{eq:1} holds with $f(n) = (4/\pi^2) \log n$ (for all $n \ge 1$). See [[#References|[a4]]], where \eqref{eq:1} is proved with $f(n) = (4/\pi^3) \log n$.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  O.C. McGehee,   L. Pigno,   B. Smith,   "Hardy's inequality and the $L^1$ norm for exponential sums"  ''Ann. of Math.'' , '''113'''  (1981)  pp. 613–618 {{ZBL|0473.42001}} {{DOI|10.2307/2007000}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  S.V. [S.V. Konyagin] Konjagin,   "On a problem of Littlewood"  ''Math. USSR Izv.'' , '''18''' :  2  (1982)  pp. 205–225  ''Izv. Akad. Nauk SSSR'' , '''45'''  (1981)  pp. 243–265</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  C.C. Graham,   O.C. McGehee,   "Essays in commutative harmonic analysis" , Springer  (1979)  pp. Chapt. 5</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.D. Stegeman,   "On the constant in the Littlewood problem"  ''Math. Ann.'' , '''261'''  (1982)  pp. 51–54</TD></TR>
</table>

====Comment====
The Littlewood problem for polynomials asks how  large the values a polynomial must be on the [[unit circle]] in  the [[complex plane]] when the coefficients of the polynomial are all $\pm 1$.  The answer to this would yield information about  the [[autocorrelation]] of binary sequences.  See [[Littlewood polynomial]].

====References====
<table>
<TR><TD valign="top">[b1]</TD> <TD valign="top">  Peter Borwein, ''Computational Excursions in Analysis and Number Theory'',  CMS Books in Mathematics '''10''', Springer (2002) {{ISBN|0-387-95444-9}} {{ZBL|1020.12001}}</TD></TR>
<TR><TD valign="top">[b2]</TD> <TD valign="top">  J.E. Littlewood, ''Some problems in real and complex analysis'', Heath Mathematical Monographs, D.C. Heath (1968) {{ZBL|0185.11502}}</TD></TR>
</table>