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In number theory, natural density, also referred to as asymptotic density or arithmetic density, is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the desired subset when combing through the interval [1, n ] as n grows large.
Although the arithmetic density is the most common way of measuring population density, several other methods have been developed to provide alternative measures of population density over a specific area. Arithmetic density: The total number of people / area of land; Physiological density: The total population / area of arable land
The logarithmic density of a set of integers from 1 to can be defined by setting the weight of each integer to be /, and dividing the total weight of the set by the th partial sum of the harmonic series (or, equivalently for the purposes of asymptotic analysis, dividing by ). The resulting number is 1 or close to 1 when the set includes all ...
Population density (people per km 2) by country. This is a list of countries and dependencies ranked by population density, sorted by inhabitants per square kilometre or square mile. The list includes sovereign states and self-governing dependent territories based upon the ISO standard ISO 3166-1.
A number N is arithmetic if the number of divisors d(N ) divides the sum of divisors σ(N ). It is known that the density of integers N obeying the stronger condition that d ( N ) 2 divides σ( N ) is 1/2.
Arable density (m² per capita) by country. This is a list of countries ordered by physiological density."Arable land" is defined by the UN's Food and Agriculture Organization, the source of "Arable land (hectares per person)" as land under temporary crops (double-cropped areas are counted once), temporary meadows for mowing or for pasture, land under market or kitchen gardens, and land ...
In arithmetic combinatorics, Szemerédi's theorem is a result concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured [1] that every set of integers A with positive natural density contains a k-term arithmetic progression for every k. Endre Szemerédi proved the conjecture in 1975.
Roth's theorem on arithmetic progressions (infinite version): A subset of the natural numbers with positive upper density contains a 3-term arithmetic progression. An alternate, more qualitative, formulation of the theorem is concerned with the maximum size of a Salem–Spencer set which is a subset of [ N ] = { 1 , … , N } {\displaystyle [N ...