Small discs containing conjugate algebraic integers

is called the transfinite diameter (or logarithmic capacity) of E. It is known that a (closed or open) disc with radius R has transfinite diameter R, whereas an interval of lenght I has transfinite diameter I/4. In [7], Fekete has shown that every compact set E satisfying τ(E) < 1 contains only finitely many full sets of conjugate algebraic integers over Q. In particular, this result can be applied to every closed disc whose radius is smaller than 1 and to every real interval whose length is smaller than 4. In the opposite direction, Fekete and Szegö [8] proved that if E is a compact set which is stable under complex conjugation and satisfies τ(E) ≥ 1, then its every complex neighborhood F (so that E ⊂ F and F is an open set) contains infinitely many sets of conjugate algebraic integers. Furthermore, by the results of Robinson [15] and Ennola [4], every real interval of length strictly greater than 4 also contains infinitely many sets of conjugate algebraic integers. In [18], Zaïmi gave a lower bound for the length of a real interval containing an algebraic integer of degree d and its conjugates. His result asserts that the length I of such an interval should be at least 4 − φ(d), where φ(d) is some explicit positive function which tends to zero as d → ∞. (For instance, one can take φ(d) = (c log d)/d with some c > 0. Similar bound also follows from an earlier result of Schur [17].) On the other hand, the author has shown that, for infinitely many d ∈ N, every real interval of length 4+4(log log d)/ log d contains an algebraic integer of degree d and its conjugates (see [2] and [3]). It is not known whether there is an interval [t, t+ 4] with some t ∈ R\Z containing infinitely many full sets of algebraic integers. For t ∈ Z, one can simply take infinitely many algebraic integers of the form t+2 cos(πr)+2, where r ∈ Q. By Kronecker’s theorem [13], these are the only such numbers in [t, t+4] if t ∈ Z.