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Results tagged with nt.number-theory
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user 8250
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
12
votes
Accepted
Density of a set of natural numbers whose differences are not bounded.
Basic definitions:
Upper and lower asymptotic density (a.k.a. natural density):
$$\overline d(A)=\limsup \frac{A(n)}n$$
$$\underline d(A)=\liminf \frac{A(n)}n$$
Upper and lower uniform density (a. …
- 4,713
3
votes
Lower density of numbers not summable by consecutive integers
Let us denote $B=\mathbb N\setminus N$, i.e., the set of consecutive-summable numbers.
There already is an answer precisely characterizing the sets $N$ and $B$. (Which is much better result than just …
- 4,713
2
votes
Lower density of {primes} times themselves
The following results are from the paper Ivan Niven, The asymptotic density of sequences. Bull. Amer. Math. Soc., 57(6):420-434, 1951. I changed the notation to denote upper asymptotic density by $\ov …
- 4,713
3
votes
Accepted
Upper density versus upper Banach density on $\omega$
I think that it is fairly straightforward to get such a set. You can simply get the set as a union of intervals:$\newcommand{\intrvl}[2]{\langle{#1},#2)}\newcommand{\intrvr}[2]{({#1},#2\rangle}\newcom …
- 4,713
23
votes
Accepted
A good reference to the general Chinese Remainder Theorem
It seems that you are after this result which can be found, for example, as Theorem 3.12 in Gareth A. Jones, Josephine M. Jones: Elementary Number Theory, Springer-Verlag, London, 1998. Springer Under …
- 4,713
4
votes
Accepted
Unicity of additive, $(-1)$-homogeneous, and shift invariant probability measures on $\mathb...
This is basically Second construction 5.3 from Eric K. van Douwen. Finitely additive measures on $\mathbb N$. Topology Appl., 47 (3), (1992), 223–268. MR1192311 (94c:28004). This issue of Topology and …
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8
votes
Accepted
Ref. request: Additive probability measure on $\mathcal P({\bf N})$ supplies subset of $\mat...
A very good reference for various forms of AC is the book Howard, Rubin: Consequences of the Axiom of Choice, AMS, 1998. (See AMS website or Google Books.) This book contains a large database of vari …
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