In this subsection we give two definitions for theories, namely simple theories and NIP, which play a significant role in the classification theory introduced by Shelah Definition 3.
Example 3. The theory of dense linear orders without end points is not simple because the formula has tree property.
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Let I be the set of rational numbers in the interval 0,1 which is a model of. For more on simplicity we refer the reader to [ 24 , Chapter 7]. It is known that stable theories are NIP. For more on the subject, cf. Note that is not a group under multiplication or addition.
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On the contrary, the theory of is very complicated. Theorem 3. Recall that the Mahler measure of an algebraic number is again an algebraic number. First we prove that the theory is not simple. In order to show this, we exhibit a formula which has the tree property. Now put Observe that for any rational number , 1 Furthermore if and are disjoint intervals of 0,1 , where , then we cannot have , otherwise we have and. Therefore, if we multiply these elements, we obtain that.
This is a contradiction, since and. For a tuple , we put. Without loss of generality, we can assume that i. This time we use the fact that the range of the logarithmic height function is dense in the positive reals.
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Again we set We shall show that this formula has the tree property by finding some parameters in some model of. Then the logarithmic height function extends to and it takes values in positive hyperreal numbers. We also denote this extension as h. Then the pair is an elementary extension of in. Note that is the set of hyperalgebraic numbers whose heights are less than 1.
Let st a denote the standard part of a finite hyperreal number. In particular, there is a hyperrational number q such that and. Observe again that for any rational number , we have. Furthermore if and are disjoint intervals of 0,1 , where , then we cannot have otherwise we have and. Similarly this is a contradiction, as and imply. Now we show that has the independence property. For simplicity, we assume that. Let be the formula. We shall show that this formula has IP. Let be given. Let and q be distinct prime numbers. Put for , where will be chosen later.
Observe that where is a positive integer and are all pairwise coprime integers, as they are distinct prime powers.
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Choose also such that 2 We put where are integers and w and are coprime. First suppose that holds, that is to say. This is a contradiction and hence i must be in I.
Conversely suppose that i is in I. This conjecture stands still open. The best known result is due to Dobrowolski 5 in , who proved that for , where. A polynomial of degree d is called reciprocal if. Thus the remaining and the difficult case for Lehmer's conjecture is the set of reciprocal algebraic numbers. The next definition plays a central role in Lehmer's conjecture. Definition 4. So a Salem number is reciprocal. It is an open question whether 1 is a limit point of Salem numbers. This is a special case of Lehmer's conjecture. The list of small Salem numbers can be found on Mossinghoff's website Small Salem numbers For more about Lehmer's conjecture and Salem numbers, we direct the reader to 21 , The rest of the paper is devoted to Lehmer's conjecture via model theory.
Recall that and where.
Note that is the set of Salem numbers. Lehmer's conjecture for Salem numbers is equivalent of there exists such that. Theorem 4. Denote the integer part of a as [ a ]. Now the pair is an elementary extension of in. Note that is the set of hyperalgebraic numbers a such that , and we have. Then there is a nonstandard number in which is infinitely close to 1, and an infinite nonstandard natural number N such that and.
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