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The Littlewood problem for compatible Diophantine approximations is the question of the existence, for any real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l0597401.png" />, of a natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l0597402.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l0597403.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l0597404.png" /> is the distance from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l0597405.png" /> to the nearest integer. In certain cases, for example for rational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l0597406.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l0597407.png" />, and for numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l0597408.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l0597409.png" /> one of which can be represented by a [[Continued fraction|continued fraction]] with non-negative elements, the Littlewood problem has an affirmative answer.
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{{TEX|part}}
  
The Littlewood problem for integrals is the problem whether for an arbitrary increasing sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l05974010.png" /> of natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l05974011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l05974012.png" /> one has
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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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l05974013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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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
 
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\begin{equation}\label{eq:1}
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l05974014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l05974015.png" /> is an absolute constant and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l05974016.png" />. The following estimates have been obtained: either weaker estimates in comparison with (*) for arbitrary sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l05974017.png" />, or estimates close to (*) or even coinciding with this estimate, but for special sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l05974018.png" />.
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\int_0^1 \left\vert { \sum_{k=1}^n \exp(2\pi i m_k x) }\right\vert dx > f(n)
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\end{equation}
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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]]]).
 
The Littlewood problems were stated by J.E. Littlewood (see [[#References|[1]]]).
  
 
====References====
 
====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>
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<table>
 
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<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>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  J.W.S. Cassels,  "An introduction to the geometry of numbers" , Springer  (1959)</TD></TR>
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</table>
  
  

Revision as of 18:28, 12 December 2014


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 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 [1]).

References

[1] G.H. Hardy, J.E. Littlewood, "A new proof of a theorem on rearrangements" J. London Math. Soc. , 23 (1948) pp. 163–168
[2] J.W.S. Cassels, "An introduction to the geometry of numbers" , Springer (1959)


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 [a1], and, independently, by S.V. Konyagin [a2]. A description of the problem just prior to its solution is given in [a3], Sect. 1.3.

For and the left-hand side of (*) is equal to the Lebesgue constant (see Lebesgue constants). As , where the are bounded and positive, it follows that the constant cannot be taken larger than . A remaining conjecture is that (*) holds with (for all ). See [a4], where (*) is proved with .

References

[a1] O.C. McGehee, L. Pigno, B. Smith, "Hardy's inequality and the norm for exponential sums" Ann. of Math. , 113 (1981) pp. 613–618
[a2] 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
[a3] C.C. Graham, O.C. McGehee, "Essays in commutative harmonic analysis" , Springer (1979) pp. Chapt. 5
[a4] J.D. Stegeman, "On the constant in the Littlewood problem" Math. Ann. , 261 (1982) pp. 51–54
How to Cite This Entry:
Littlewood problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Littlewood_problem&oldid=15076
This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article