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In this paper we show that for eachk∈ ℕ there are infinitely many algebraic integers with normkand absolute normalized size smaller than 1. We also show that the lower bound (n+slog 2)/2 on the square of the absolute size ‖α‖ of an algebraic integer α of degreenwith exactlysreal conjugates over ℚ is best possible for each evens> 2. For this, for each pairs,k∈ ℕ, wheresis even, we construct algebraic integers α with exactlysreal conjugates and norm of modulusksatisfying deg α =nand ‖α‖2= (n+slog 2)/2 + logk+O(n−1) asn→ ∞ to. Finally, using the third smallest Pisot number θ3, which is the root of the polynomialx5–x4–x3+x2–1, we construct algebraic integers α of degreenthat have exactly one real conjugate and satisfy ‖α‖2≤ n/2 + 0.346981 … (which is quite close to the above lower bound (n+ log 2)/2 = n/2 + 0.346573 … fors= 1). In the proofs we use some irreducibility theorems for lacunary polynomials and the Erdős and Turán bound on the number of roots of a polynomial in a sector. [ABSTRACT FROM AUTHOR]