The ternary Goldbach problem with restrictions on the summands
Using the combinatorial methods of \textit{P. Erdős} [Proc. Am. Math. Soc. 5, 847-853 (1954; Zbl 0056.270)], and \textit{G. Dufner} and \textit{D. Wolke} [Monatsh. Math. 118, 75-82 (1994; Zbl 0808.11060)] which exploit the fact that every large odd integer has many representations as a sum of three primes, \textit{E. Wirsing}'s result on thin bases of primes for the set of positive integers [Analysis 6, 285-308 (1986; Zbl 0586.10032)] is deduced from Vinogradov's three primes theorem and slightly generalized. The author proves that if \(\alpha_ j>0\) for \(j=1,2,3\), \(\alpha_ 1+ \alpha_ 2+ \alpha_ 3= 1\), and \(\beta_ 1+ \beta_ 2+ \beta_ 3=1\) then there exist subsets \(\mathbb{P}_ j\) of the set of all prime numbers \(\mathbb{P}\), such that \(| \mathbb{P}_ j\cap [1,x]| \ll x^{\alpha_ j} (\log x)^{\beta_ j}\), and every large odd number can be represented as \(p_ 1+ p_ 2+ p_ 3\), with \(p_ j\in \mathbb{P}_ j\). The author remarks that one can take \(\alpha_ 1=0\), provided that \(\beta_ 1\geq 2\). It is also possible to have two very thin sets of primes \(\mathbb{P}_ 1\) and \(\mathbb{P}_ 2\) (with \(\alpha_ 1= \alpha_ 2 =0\) and \(\beta_ 1= \beta_ 2 =1\)), at the cost of taking \(\mathbb{P}_ 3= \mathbb{P}\).
scientific article; zbMATH DE number 624941