The size of algebraic integers with many real conjugates

In this paper we show that the relative normalised size with respect to a number field K of an algebraic integer α  ≠ -1, 0, 1 is greater than 1 provided that the number of real embeddings s of K satisfies s ≥ 0.828 n , where n = [ K : Q ]. This can be compared with the previous much more restrictive estimate s ≥ n  − 0.192 √ ( n /log n ) and shows that the minimum m ( K ) over the relative normalised size of nonzero algebraic integers α  in such a field K is equal to 1 which is attained at α  = ± 1. Stronger than previous but apparently not optimal bound for m ( K ) is also obtained for the fields K satisfying 0.639 ≤ s / n < 0.827469…. In the proof we use a lower bound for the Mahler measure of an algebraic number with many real conjugates.