The distribution of non-Parry Perron numbers and their conjugates

Abstract We prove that the set of Perron numbers that are not Parry numbers is dense in the interval $$(1,\infty )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and that the set of all Galois conjugates of such numbers is dense in the whole complex plane. While the Parry numbers themselves are known also to be dense in $$(1,\infty )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , their conjugates are known from work of Solomyak and others to be much more restricted: they are confined to a subset of the disc $$|z|&lt;(1+\sqrt{5})/2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>z</mml:mi> <mml:mo>|</mml:mo> <mml:mo>&lt;</mml:mo> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:msqrt> <mml:mn>5</mml:mn> </mml:msqrt> <mml:mo>)</mml:mo> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . Our work is in response to a remark of Akiyama, drawing attention to our lack of knowledge about non-Parry Perron numbers and their conjugates.