Classical and uniform exponents of multiplicative p -adic approximation
Let p be a prime number and ξ an irrational p-adic number. Its irrationality exponent µ(ξ) is the supremum of the real numbers µ for which the system of inequalities 0 < max{|x|, |y|} ≤ X, |yξ - x|p ≤ X-µ has a solution in integers x, y for arbitrarily large real number X. Its multiplicative irratio...
| Authors: | ; |
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| Format: | Article |
| Publication date: | 2024 |
| Country: | España |
| Institution: | Universitat Autònoma de Barcelona |
| Repository: | Dipòsit Digital de Documents de la UAB |
| Language: | English |
| OAI Identifier: | oai:ddd.uab.cat:286796 |
| Online Access: | https://ddd.uab.cat/record/286796 https://dx.doi.org/urn:doi:10.5565/PUBLMAT6812401 |
| Access Level: | Open access |
| Keyword: | Rational approximation P-adic number Exponent of approximation |
| Summary: | Let p be a prime number and ξ an irrational p-adic number. Its irrationality exponent µ(ξ) is the supremum of the real numbers µ for which the system of inequalities 0 < max{|x|, |y|} ≤ X, |yξ - x|p ≤ X-µ has a solution in integers x, y for arbitrarily large real number X. Its multiplicative irrationality exponent µ×(ξ) (resp., uniform multiplicative irrationality exponent µb×(ξ)) is the supremum of the real numbers µb for which the system of inequalities 0 < |xy| 1/2 ≤ X, |yξ - x|p ≤ X-µb has a solution in integers x, y for arbitrarily large (resp., for every sufficiently large) real number X. It is not difficult to show that µ(ξ) ≤ µ×(ξ) ≤ 2µ(ξ) and µb×(ξ) ≤ 4. We establish that the ratio between the multiplicative irrationality exponent µ× and the irrationality exponent µ can take any given value in [1, 2]. Furthermore, we prove that µb×(ξ) ≤ (5 + √ 5)/2 for every p-adic number ξ. |
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