On the beta-expansions of 1 and algebraic numbers for a Salem number beta

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Abstract

<jats:title>Abstract</jats:title><jats:p>We study the digits of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0143385713000990_inline1" /><jats:tex-math>$\beta $</jats:tex-math></jats:alternatives></jats:inline-formula>-expansions in the case where <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0143385713000990_inline2" /><jats:tex-math>$\beta $</jats:tex-math></jats:alternatives></jats:inline-formula> is a Salem number. We introduce new upper bounds for the numbers of occurrences of consecutive 0s in the expansion of 1. We also give lower bounds for the numbers of non-zero digits in the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0143385713000990_inline3" /><jats:tex-math>$\beta $</jats:tex-math></jats:alternatives></jats:inline-formula>-expansions of algebraic numbers. As applications, we give criteria for transcendence of the values of power series at certain algebraic points.</jats:p>

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