Additive properties of dense subsets of sifted sequences

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g_functions: Exact residue-support, uniform-lifting and G-function conventions from Section 4.1, including the comparison used in the improved large sieve.

theorem_4: The exact nonnegative decomposition of reduced-fraction Fourier energy for a sequence supported by a compact satisfying uniform lifting.

theorem_5: The two-level Fourier bound with factor G_1(Q_0)/G_1(Q/Q_0) and enlarged-level cost N+Q squared, with its exact support hypotheses.

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Olivier Ramaré and Imre Z. Ruzsa, Additive properties of dense subsets of sifted sequences, Journal de Théorie des Nombres de Bordeaux 13 (2) (2001), 559–581, DOI 10.5802/jtnb.338. The edition read is the journal scan at the Numdam record, obtained from its PDF endpoint.

Section 4.1 supplies a sieve interface that allows prime-power moduli. Its supporting compact has compatible residue sets, and the Johnsen condition requires uniform lifting between successive prime-power levels. The G-functions encode these sets. Under those hypotheses, Theorem 4 expresses Fourier energy as a nonnegative sum, and Theorem 5 bounds the energy through a ratio of G-functions at two different levels.

Read status: claims checked. The support definition (4.1.1), G-functions (4.1.3)–(4.1.6), Lemma 1, Johnsen condition (4.1.12), and Theorems 4 and 5 were checked against images of printed pp. 567–571. Their proofs on pp. 570–571 were read for the stated interfaces, without an independent proof audit. No reading or verification of the paper's other theorems is asserted.

Bears on. The two-level estimate is a possible input to the multiscale large-sieve approach to Problem 12(iii). An application must specify its actual residue sets and support, check the uniform-lifting condition, estimate both G-functions, and retain the enlarged level's quadratic cost. This source supplies no property-P guard-budget bound.

The scan's cover states that Université Bordeaux 1 reserves all rights to the 2001 work. That notice and Numdam's reuse terms were read on 2026-10-17. Numdam's permission to host the scan does not grant redistribution elsewhere. No file of this source is held because no license on record permits redistribution.