We study the problem of designing a data structure that reports the positions of the distinct τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document}-majorities within any range of an array A[1,n]\documentclass[12pt]{minimal}
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\begin{document}$$A[1,n]$$\end{document}, without storing A\documentclass[12pt]{minimal}
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\begin{document}$$A$$\end{document}. A τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document}-majority in a range A[i,j]\documentclass[12pt]{minimal}
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\begin{document}$$A[i,j]$$\end{document}, for 0<τ<1\documentclass[12pt]{minimal}
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\begin{document}$$0<\tau < 1$$\end{document}, is an element that occurs more than τ(j-i+1)\documentclass[12pt]{minimal}
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\begin{document}$$\tau (j-i+1)$$\end{document} times in A[i,j]\documentclass[12pt]{minimal}
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\begin{document}$$A[i,j]$$\end{document}. We show that Ω(n⌈log(1/τ)⌉)\documentclass[12pt]{minimal}
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\begin{document}$$\Omega (n\lceil \log (1/\tau )\rceil )$$\end{document} bits are necessary for any data structure just able to count the number of distinct τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document}-majorities in any range. Then, we design a structure using O(n⌈log(1/τ)⌉)\documentclass[12pt]{minimal}
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\begin{document}$$O(n\lceil \log (1/\tau )\rceil )$$\end{document} bits that returns one position of each τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document}-majority of A[i,j]\documentclass[12pt]{minimal}
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\begin{document}$$A[i,j]$$\end{document} in O((1/τ)loglogw(1/τ)logn)\documentclass[12pt]{minimal}
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\begin{document}$$O((1/\tau )\log \log _w(1/\tau )\log n)$$\end{document} time, on a RAM machine with word size w\documentclass[12pt]{minimal}
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\begin{document}$$w$$\end{document} (it can output any further position where each τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document}-majority occurs in O(1)\documentclass[12pt]{minimal}
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\begin{document}$$O(1)$$\end{document} additional time). Finally, we show how to remove a logn\documentclass[12pt]{minimal}
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\begin{document}$$\log n$$\end{document} factor from the time by adding O(nloglogn)\documentclass[12pt]{minimal}
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\begin{document}$$O(n\log \log n)$$\end{document} bits of space to the structure.
