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Complete sequences

A sequence of positive integers typeset structure is called to be complete if every sufficiently large positive integer typeset structure can be written as a sum of distinct terms of typeset structure, that is if the set

Σ(A) = {Underoverscript[∑, i = 1, arg3] ε _ i a _ i :     &e ...  i ∈ {0, 1},    Underoverscript[∑, i = 1, arg3] ε _ i < ∞ }(1)

contains all sufficiently large integers.

Let typeset structure denote the number of elements typeset structure with typeset structure.

We have several general sufficient conditions for completeness:

Theorem ( [1] ). Let typeset structure be a sequence of distinct integers. Let typeset structure. Suppose that

Underscript[lim, n -> ∞] (A(2 n) - A(n))/(log log n) = ∞(2)

and that

Underscript[∑, n] || θ a _ n ||^2 = ∞(3)

for every real number typeset structure, typeset structure. Then typeset structure is complete.

Corolarry ( [2]  ). Let typeset structure, typeset structure, be a polynomial with integral coefficients such that for every integer typeset structure there is an infinity of primes typeset structure with typeset structure. Then the sequence typeset structure, where typeset structure runs through the primes, is complete.

Theorem  (  [3] ). Let typeset structure be a sequence of positive integers satisfying

(1) typeset structure for all typeset structure where typeset structure,

(2) for every typeset structure, each residue class modulo typeset structure contains at least one integer which is a sum of distinct typeset structures.

Then typeset structure is complete.

Corolarry  ( [4]  ). Let typeset structure be a sequence of positive integers for which

(1) for each typeset structure there exists an typeset structure such that typeset structure,

(2) for every integer typeset structure there are only finitely many terms in typeset structure divisible bt typeset structure.

Let typeset structure, typeset structure, denote the sequence of those terms from typeset structure whose subscripts are in typeset structure. Then typeset structure is complete for every typeset structure.

Theorem (  [5] ). Let typeset structure be an infinite sequence of positive integers satisfying

(1) typeset structure for typeset structure,

(2) at least one element of every arithmetic progression is expressible as a sum of distinct terms of typeset structure.

Then typeset structure is complete.

Birch  [6] answered in affirmative way a problem (cf. the Corolarry) posed by P.Erdös proving

Theorem. Let typeset structure be coprime positive integers, and   typeset structure where typeset structure and typeset structure are arbitrary and typeset structure is sufficiently large. Then typeset structure is complete.

Corollary. If typeset structure are coprime positive integers, then the sequence typeset structure is complete.

References

[1]  Cassels, J. W. (1960). On the representation of integers as the sums of distinct summands taken from a fixed set. Acta Sci. Math., 21, 111-124.

[2]  Porubský, Š. (1979). Sums of prime powers. Monatshefte für Mathematik, 86, 301-303.

[3]  Folkman, J. (1966). On the representation of integers as sums of distinct terms from a fixed sequence. Can. J. Math., 18, 643-655.

[4]  Porubský, Š. (1978). Sums of distinct terms from a fixed sequence. Nordisk. Mat. Tidskr., 25-26, 185-187.

[5]  Erdös, P. (1962). On the representation of large integers as sums of distinct summands taken from a fixed set. Acta Arith., 7, 345-354.

[6]  Birch, B. J. (1959). Note on a problem of Erdös. Proc. Camb. Philos. Soc., 55, 370-373.

Cite this web-page as:

Štefan Porubský: Complete sequence.

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