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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|done}}
  
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>
 +
<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>
  
  
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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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l05974019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l05974020.png" /> the left-hand side of (*) is equal to the Lebesgue constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l05974021.png" /> (see [[Lebesgue constants|Lebesgue constants]]). As <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l05974022.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l05974023.png" /> are bounded and positive, it follows that the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l05974024.png" /> cannot be taken larger than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l05974025.png" />. A remaining conjecture is that (*) holds with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l05974026.png" /> (for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l05974027.png" />). See [[#References|[a4]]], where (*) is proved with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l05974028.png" />.
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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>
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<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>
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<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>
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</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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  O.C. McGehee,   L. Pigno,  B. Smith,  "Hardy's inequality and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059740/l05974029.png" /> norm for exponential sums"  ''Ann. of Math.'' , '''113''' (1981) pp. 613–618</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>
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<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>

Latest revision as of 19:43, 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 $N=2n+1$ and $m_k = k$ the left-hand side of \eqref{eq:1} is equal to the 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 [a4], where \eqref{eq:1} is proved with $f(n) = (4/\pi^3) \log n$.

References

[a1] 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
[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

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

[b1] Peter Borwein, Computational Excursions in Analysis and Number Theory, CMS Books in Mathematics 10, Springer (2002) ISBN 0-387-95444-9 Zbl 1020.12001
[b2] J.E. Littlewood, Some problems in real and complex analysis, Heath Mathematical Monographs, D.C. Heath (1968) Zbl 0185.11502
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