Essential Components
Khintchin proved that the sequence of squares, though of zero Schnirelmann density, when added to a sequence of Schnirelmann density between 0 and 1, increases the density:
This was soon simplified and extended by Erdős, who showed, that if A is any sequence with Schnirelmann density α and B is an additive basis of order k then
Sequences with this property, of increasing density less than one by addition, were named essential components by Khintchin. Linnik showed that an essential component need not be an additive basis as he constructed an essential component that has xo(1) elements less than x. More precisely, the sequence has
elements less than x for some c < 1. This was improved by E. Wirsing to
For a while, it remained an open problem how many elements an essential component must have. Finally, Ruzsa determined that an essential component has at least (log x)c elements up to x, for some c > 1, and for every c > 1 there is an essential component which has at most (log x)c elements up to x.
Read more about this topic: Schnirelmann Density
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