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  "body": "\n## Statement\n\nLet $(\\lambda_k)_{k \\ge 1}$ be a strictly increasing sequence of nonnegative integers such that $\\lambda_k / k \\to \\infty$ as $k \\to \\infty$.\nLet $(a_k)$ be complex numbers such that the [[power series|0VP56B4EMX]]\n$$f(z) = \\sum_{k=1}^{\\infty} a_k z^{\\lambda_k}$$\nhas a [[radius of convergence|0660QX1VPR]] $R$ with $0 \u003c R \u003c \\infty$.\nThen the circle $\\abs{z} = R$ is a [[natural boundary|PQFEA12AHA]] of $f$.\nThat is, $f$ has no [[analytic continuation|0M3YQJ81NB]] to a neighbourhood of any point of this circle.\n\n## Remarks\n\nThe condition $\\lambda_k / k \\to \\infty$ says that the exponents that occur have density zero among the nonnegative integers.\nIt is weaker than the condition $\\lambda_{k+1} \\ge q \\lambda_k$ with $q \u003e 1$ of the [[Hadamard gap theorem|DA8M8VZZHC]], so the Fabry theorem contains the Hadamard theorem.\nFor example, the series $\\sum_{k \\ge 1} z^{k^2}$ has radius $1$, and $k^2 / k = k \\to \\infty$, so the unit circle is a natural boundary; the Hadamard theorem does not apply here, because $(k+1)^2 / k^2 \\to 1$.\nThe gap condition cannot simply be dropped: the series $\\sum_{k \\ge 0} z^k = 1/(1-z)$ has radius $1$ and continues analytically across every point of the unit circle except $z = 1$.\nThe proof is harder than for the Hadamard theorem and is not sketched here.\n\n## Explanations\n\n### L1 human\n\nA power series with very many missing powers cannot be extended beyond its disc of convergence at all.\nEach point of the boundary circle is a singular point.\n\"Very many missing\" means that among the first $N$ powers only a vanishing proportion appear.\nThe series $z + z^4 + z^9 + z^{16} + \\cdots$ is an example.\n\n### L2 human\n\nUsually a power series has only a few singular points on its circle of convergence, and the function continues across the rest.\nGap theorems show the opposite behaviour for lacunary series.\nThe density condition on the exponents is the essential hypothesis, and it is much weaker than a fixed ratio between consecutive exponents.\nIt shows that lacunary series are a simple source of functions with a natural boundary.\n",
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