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  "body": "\n## Definition\n\nFor a real number $x$, write $e(x) = e^{2\\pi i x}$, where $e^{z}$ is the [[complex exponential|07GJV36896]].\nLet $A$ be a finite set of integers, and let $f \\colon A \\to \\R$ be a function.\nThe *exponential sum* of $f$ over $A$ is the complex number\n$$S = \\sum_{n \\in A} e(f(n)).$$\nMost often $A$ is an interval $\\set{n \\in \\Z : M \u003c n \\le M + N}$.\n\n## Remarks\n\nEach term has absolute value $1$, so $\\abs{S} \\le \\#A$; this is the trivial bound.\nExample: for $f(n) = n\\alpha$ with $\\alpha \\in \\R \\setminus \\Z$ and $A = \\set{1, \\dots, N}$, the sum is geometric and\n$$\\abs{S} = \\abs{\\frac{e(N\\alpha) - 1}{e(\\alpha) - 1}} \\le \\frac{1}{\\abs{\\sin \\pi \\alpha}},$$\na bound independent of $N$.\nNon-example: $\\sum_{n=1}^{N} 2^n$ is a sum of values of an exponential function, but it is not an exponential sum in this sense.\nIts terms have absolute value larger than $1$, so they are not of the form $e(f(n))$ with $f$ real.\nWhen $f$ is a polynomial, $S$ is called a Weyl sum.\nSums with weights, $\\sum_{n \\in A} a_n e(f(n))$ with complex $a_n$, are also called exponential sums.\n",
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