# Littlewood problem

The Littlewood problem for compatible Diophantine approximations is the question of the existence, for any real numbers , of a natural number such that , where is the distance from to the nearest integer. In certain cases, for example for rational and , and for numbers and 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 of natural numbers , one has

(*) |

with , where is an absolute constant and . The following estimates have been obtained: either weaker estimates in comparison with (*) for arbitrary sequences , or estimates close to (*) or even coinciding with this estimate, but for special sequences .

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 |

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Littlewood problem.

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