A study of quasi-integers by FERNANDA CARES CUEVAS

What are quasi-integral points?

facares — 2022-10-25 02:24:58 UTC

\text{ Let } \alpha=\frac{a}{b},

where a,b are coprime integers.

\text{ Let } 0 \leq r\leq 1,

\text{ then } \alpha \text{ is r-quasi-integral if }

|b|\leq \max\{|a|,|b|\}^r.

Example 1:

facares — 2022-10-25 02:38:02 UTC

-\frac{4}{3} \text{ is a } 0.8-\text{quasi-integer because }

4^{0.8} = 4^{4/5} = (256)^{1/5} > 243^{1/5} = 3.

Example 2:

facares — 2022-10-25 02:42:54 UTC

\frac{31}{12} \text{ is a } 0.8-\text{quasi-integer because }

31^{0.8} > 15,59 > 12.

Example 3:

facares — 2022-10-25 02:46:49 UTC

-\frac{5}{4} \text{ is not a } 0.8-\text{quasi-integer because }

5^{0.8} = 5^{4/5} = 25^{2/5} < 32^{2/5} = 2^2 = 4

Relationship with integers

facares — 2022-10-25 03:00:23 UTC

If we study 0-quasi-integers, we note that they are the fractions a/b such that

|b|\leq \max\{|a|,|b|\}^0=1.

As b is a non zero integer,

|b|=1 \rightarrow b=\pm 1.

Then the fraction a/b was an integer number!

Our objetive

facares — 2022-10-25 03:08:14 UTC

To see the structure that quasi-integers preserves and changes contrasting with integers.
In particular, we will generalize Siegel's Theorem to the quasi-integral case.

Another extreme case

facares — 2022-10-25 03:16:02 UTC

If we study 1-quasi-integers, they are irreductible fractions a/b such that

|b| \leq \max\{|a|,|b|\}^1

and this is true always! So every rational number is a 1-quasi-integer.

Integers

facares — 2022-10-25 03:21:39 UTC

Integers are closed by the sum, this means that if we sum two integers if will result in a integer.

Quasi-integers

facares — 2022-10-25 03:24:40 UTC

Not for every r the set of r-quasi-integers is closed by the sum. For example:

\frac{5}{4} = \frac{31}{12} + \frac{-4}{3},

and the numbers in the right are 0.8-quasi-integers but 5/4 is not!

Integers

facares — 2022-10-25 03:30:17 UTC

Units are the numbers such that its multiplicative inverse is an integer.
In this case, this numbers are 1 and -1.

Quasi-integers

facares — 2022-10-25 03:31:14 UTC

Let r<1, if a/b is a r-quasi-integers, could it be b/a a r-quasi-integer too?

Well, this is false for ever number that is not 1 and -1. Because every number x with

|x|<1

is not a r-quasi-integer for every r<1.

So the units of r-quasi-integers are 1,-1 when r<1.
When r=1, every non zero number is integer.

Increasing sets

facares — 2022-10-25 03:37:29 UTC

By the definition, if r
~~The set of r-quasi-integral numbers are increasing with r from integers (r=0) to all the rational numbers (r=1).~~

Using in another areas

facares — 2022-10-25 03:53:44 UTC

With our generalizations we could try to prove some theorems of arithmetic dynamics, another area of mathematics, for r-quasi-integers.

Working with another notion of integers

facares — 2022-10-25 03:58:02 UTC

There is another notion of integer: the S-integers, where S is a set of absolute values. This generalization of integral numbers adds conditions over the denominator of the fraction depending on a prime. For example,
\frac{2}{3} \text{ is a } \{\infty,3\}-\text{integral point}.
Then we can generalize the notion to a quasi-(r,S)-integer and try to prove things there.

Working of finite extentions of rational numbers

facares — 2022-10-25 04:00:27 UTC

Every definition could be extended to some sets greater than the rational numbers but with a field structure.
We could try to understand the structure in this case and see if a generalization of Siegel's Theorem is possible.

Integers

facares — 2022-10-27 01:58:06 UTC

Let
G(x,y)=\frac{x^3}{x^3-y^3},
Siegel's theorem implies that there are only finitely many rationals (x,y) such that G(x,y) is an integer.

Siegel's theorem can be applied in a funtion that is a division of two polynomials with the condicions:
The coefficients of the polynomials have to be rationals
The function on the denominator has al least 3 complex solutions.
The polynomial in the numerator and denominator are homogeneous of the same degree (this is, every term has the same degree in both of them).

Quasi-integers

facares — 2022-10-27 02:01:30 UTC

As a result of our investigation, we proved a generalization of Siegel theorem for r-quasi-integers: With the same conditions as above, there is a constant between 0 and 1: c such that there are only finitely rationals such that its image is a c-quasi-integer.

facares — 2022-10-27 02:06:21 UTC

M. Hindry and J. H. Silverman. Diophantine geometry. An introduction. Graduate Texts in Mathematics, 201. Springer-Verlag, New York, 2000.
L. Hsia and J. H. Silverman. A quantitative estimate for quasiintegral points in
orbits. Pacific J. Math. 249 (2011), no. 2, 321–342.
J. H. Silverman. Integer points, Diophantine approximation, and iteration of rational
maps. Duke Math. J. 71 (1993), no. 3, 793–829.