ON THE FUNCTION $ G(n)$ IN WARING'S PROBLEM

Карацуба, Анатолий Алексеевич

doi:10.1070/im1986v027n02abeh001176

Cited by 7 publications

(4 citation statements)

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“…The proof of Theorem 2 uses an estimate for certain exponential sums which is obtained by a technique motivated by Vinogradov's treatment [8] of the sum X « "• However, in place of the Vinogradov Mean Value Theorem, we use a 1-dimensional analogue, following an idea of Karatsuba [5]. This entails restricting the values of n which occur in Theorem 2 to those which are well-factorable, in an appropriate sense.…”

Section: Let a Eu And E >0 Be Given For Any Integer Ks»6 There Are Imentioning

confidence: 99%

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The fractional part of αn ^k

Heath‐Brown

1988

Mathematika

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Section: Let a Eu And E >0 Be Given For Any Integer Ks»6 There Are Imentioning

confidence: 99%

“…Proof of Theorem 2: A Mean Value Theorem. The ideas of this section are essentially contained in the work of Karatsuba [5]. We begin by defining the weights a,,..., a N which we shall use in (2.2).…”

Section: (Aq) = Lmentioning

confidence: 99%

The fractional part of αn ^k

Heath‐Brown

1988

Mathematika

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“…Let G(k) = G N (k) be the smallest number such that for all s ≥ G(k), every large enough natural number can be written as a sum of s positive integral k-th powers. Vinogradov [16], Karatsuba [4] and Vaughan [14] made progress to achieve upper bounds for G(k), the best current one for large k being…”

Section: Introductionmentioning

confidence: 99%

Uniform bounds in Waring's problem over some diagonal forms

Pliego¹

2020

Preprint

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“…The question resulted from the analysis of the Waxing problem undertaken by the author in [2][3][4][5].…”

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confidence: 99%

Regular sets in residual classes

Карацуба

1998

Math Notes

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ABSTRACT. Sets regular modulo a fixed odd prime power are explicitly constructed under the condition that their cardinalities do not exceed an arbitrarily small positive power of the modulus.KEY WORDS: regular sets modulo a prime number, Waxing problem, additive congruences.In [I] the question was raised as to whether there exist regular sets modulo a given number m whose cardinalities do not exceed m ~ , where 0 < e < 1, m _> rot(e), and, for m = pN with a fixed odd prime p, such sets were constructed explicitly. The question resulted from the analysis of the Waxing problem undertaken by the author in [2][3][4][5].Suppose that m is a positive integer, m _> ml > 0, Em is a complete residue system modulo m, A C Era, and IIAII is the cardinality of A. The regular sets A in this theorem consist of the modulo m residues of the numbers z", z = 1, 2,..., [m~], (x, m) = 1, where n is an arbitrary prime from the interval (32e -2 , 64e-2].In a more general setting, a similar problem was considered by Nathanson in [6], where the history of the problem (the Rohrbach problem for finite groups) is described and the following theorem is proved.

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ON THE FUNCTION $ G(n)$ IN WARING'S PROBLEM

Cited by 7 publications

References 1 publication

The fractional part of αn ^k

The fractional part of αn ^k

Uniform bounds in Waring's problem over some diagonal forms

Regular sets in residual classes

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ON THE FUNCTION $ G(n)$ IN WARING'S PROBLEM

Cited by 7 publications

References 1 publication

The fractional part of αn k

The fractional part of αn k

Uniform bounds in Waring's problem over some diagonal forms

Regular sets in residual classes

Contact Info

Product

Resources

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The fractional part of αn ^k

The fractional part of αn ^k