APPROXIMATION BY ALGEBRAIC NUMBERS

By M. KAENA

CONTENTS APPROXIMATION BY RATIONAL NUMBERS DIRICHLET AND LIOUVILLE CONTINUED FRACTIONS THE THEOREM OF KHINTCHINE THE DUF?N–SCHAEFFER CONJECTURE COMPLEMENTARY RESULTS ON CONTINUED FRACTIONS EXERCISES NOTES APPROXIMATION TO ALGEBRAIC NUMBERS RATIONAL APPROXIMATION EFFECTIVE RATIONAL APPROXIMATION APPROXIMATION BY ALGEBRAIC NUMBERS EFFECTIVE APPROXIMATION BY ALGEBRAIC NUMBERS REMARKS ON IRRATIONALITY AND TRANSCENDENCE STATEMENTS THE CLASSI?CATIONS OF MAHLER AND KOKSMA MAHLER’S CLASSI?CATION SOME PROPERTIES OF MAHLER’S CLASSI?CATION KOKSMA’S CLASSI?CATION COMPARISON BETWEEN BOTH CLASSI?CATIONS SOME EXAMPLES EXPONENTS OF DIOPHANTINE APPROXIMATION MAHLER’S CONJECTURE ON S-NUMBERS STATEMENTS OF THE THEOREMS AN AUXILIARY RESULT EXERCISE HAUSDORFF DIMENSION OF EXCEPTIONAL SETS HAUSDORFF MEASURE AND HAUSDORFF DIMENSION UPPER BOUND FOR THE HAUSDORFF DIMENSION THE MASS DISTRIBUTION PRINCIPLE REGULAR SYSTEMS THE THEOREM OF JARN´IK–BESICOVITCH HAUSDORFF DIMENSION OF SETS OF S?-NUMBERS HAUSDORFF DIMENSION OF SETS OF S-NUMBERS RESTRICTED DIOPHANTINE APPROXIMATION 5 10 NOTES DEEPER RESULTS ON THE MEASURE OF EXCEPTIONAL SETS OPTIMAL REGULAR SYSTEMS A KHINTCHINE-TYPE RESULT HAUSDORFF MEASURE OF EXCEPTIONAL SETS SETS WITH LARGE INTERSECTION PROPERTIES APPLICATION TO THE APPROXIMATION BY ALGEBRAIC NUMBERS ON T-NUMBERS AND U-NUMBERS T -NUMBERS DO EXIST THE INDUCTIVE CONSTRUCTION ON THE GAP BETWEEN W? N AND WN HAUSDORFF DIMENSION AND HAUSDORFF MEASURE ON U-NUMBERS A METHOD OF G¨UTING BRIEF SUMMARY OF THE RESULTS TOWARDS THE MAIN PROBLEM 7 10 NOTES OTHER CLASSI?CATIONS OF REAL AND COMPLEX NUMBERS SPRIND?ZUK’S CLASSI?CATION ANOTHER CLASSI?CATION PROPOSED BY MAHLER TRANSCENDENCE MEASURES AND MEASURES OF ALGEBRAIC APPROXIMATION APPROXIMATION IN OTHER ?ELDS APPROXIMATION IN THE ?ELD OF COMPLEX NUMBERS APPROXIMATION IN THE ?ELD OF GAUSSIAN INTEGERS APPROXIMATION IN THE P-ADIC ?ELDS APPROXIMATION IN ?ELDS OF FORMAL POWER SERIES CONJECTURES AND OPEN QUESTIONS 10 1 THE LITTLEWOOD CONJECTURE 10 2 OPEN QUESTIONS 10 3 NOTES REFERENCES