Schnirelmann Density - Essential Components

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

Famous quotes containing the words essential and/or components:

    The memory loaded with mere bookwork is not the thing wanted—is, in fact, rather worse than useless—in the teacher of scientific subjects. It is absolutely essential that his mind should be full of knowledge and not of mere learning, and that what he knows should have been learned in the laboratory rather than in the library.
    Thomas Henry Huxley (1825–95)

    Hence, a generative grammar must be a system of rules that can iterate to generate an indefinitely large number of structures. This system of rules can be analyzed into the three major components of a generative grammar: the syntactic, phonological, and semantic components.
    Noam Chomsky (b. 1928)