On a Problem of Erdös in Additive Number Theory Original Research Article

In 1954 Lorentz and Erdös showed that there are very thin sets of positive integers complementary to the set of primes. In particular, there is anAsubset of or equal toimage with[formula]and[formula]Erdös conjectured that the bound (i) could be sharpened too(ln2 x) or evenO(ln x). In the present paper it is proved that there are setsAwith[formula]such that (ii) is fulfilled for almost alln.