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Displaying similar documents to “Limitation to the asymptotic formula in Waring's problem”

On normal lattice configurations and simultaneously normal numbers

Mordechay B. Levin (2001)

Journal de théorie des nombres de Bordeaux

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Let q , q 1 , ⋯ , q s ≥ 2 be integers, and let α 1 , α 2 , ⋯ be a sequence of real numbers. In this paper we prove that the lower bound of the discrepancy of the double sequence ( α m q n , ⋯ , α m + s - 1 q n ) m , n = 1 M N coincides (up to a logarithmic factor) with the lower bound of the discrepancy of ordinary sequences ( x n ) n = 1 M N in s -dimensional unit cube ( s , M , N = 1 , 2 , ⋯ ) . We also find a lower bound of the discrepancy (up to a logarithmic factor) of the sequence ( α 1 q 1 n , ⋯ , α s q s n ) n = 1 N (Korobov’s problem).

On partitions without small parts

J.-L. Nicolas, A. Sárközy (2000)

Journal de théorie des nombres de Bordeaux

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Let r ( n , m ) denote the number of partitions of n into parts, each of which is at least m . By applying the saddle point method to the generating series, an asymptotic estimate is given for r ( n , m ) , which holds for n → ∞ , and 1 ≤ m ≤ c 1 n log n c 2 .