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  "body": "\n## Definition\n\nA strictly increasing sequence $(n_k)_{k \\ge 1}$ of positive integers is *lacunary* (in the sense of Hadamard) if there is a number $q \u003e 1$ with\n$$\\frac{n_{k+1}}{n_k} \\ge q \\quad \\text{for all } k \\ge 1.$$\nA *lacunary trigonometric series* is a [[trigonometric series|FZK2M8KGXQ]] of the form\n$$\\sum_{k \\ge 1} \\left( a_k \\cos n_k x + b_k \\sin n_k x \\right),$$\nwith real or complex coefficients $a_k, b_k$ and a lacunary sequence $(n_k)$.\nA *lacunary power series* is a [[power series|0D6KXBHKC0]] $\\sum_{k \\ge 1} c_k z^{n_k}$ with a lacunary sequence $(n_k)$.\n\n## Remarks\n\nExample: $\\sum_{k \\ge 0} 2^{-k} \\cos(2^k x)$ is lacunary with $q = 2$, since $2^{k+1}/2^k = 2$.\n\nNon-example: $\\sum_{k \\ge 1} \\frac{\\cos(k^2 x)}{k^2}$ has frequencies $n_k = k^2$ with $n_{k+1}/n_k = (1 + 1/k)^2 \\to 1$.\nSo no $q \u003e 1$ is a lower bound for all ratios, and the series is not lacunary in the sense of Hadamard.\n\nA theorem of Sidon says that if a lacunary trigonometric series is the Fourier series of a bounded function, then $\\sum_k (\\abs{a_k} + \\abs{b_k}) \u003c \\infty$.\nFor power series the [[Hadamard gap theorem|DA8M8VZZHC]] applies.\nWeaker gap conditions, such as $n_k / k \\to \\infty$ in the [[Fabry gap theorem|GE0HX2KMMZ]], are also studied.\n",
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