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        "description": "A qualitative asymptotic theorem for each sequence or fixed polynomial under the stated irrationality conditions, not a uniform numerical discrepancy rate over all irrational coefficients or a finite-sample randomness guarantee. An irrational constant term alone is insufficient. The scalar linear case predates this paper and is credited jointly in the original to Bohl, Sierpiński and Weyl; Hardy–Littlewood's earlier and parallel polynomial work is explicitly retained. Later metric theorems and the geometric appendix lie outside this record.",
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      "statement": "# Weyl: testing equidistribution by exponential sums and applying it to polynomials\n\n## A pattern can look evenly scattered\n\nWalk around a circle by the same irrational fraction of a turn each time and mark your positions. No random-number generator is involved. Nevertheless, over the long run an arc occupying a tenth of the circle receives a tenth of the marks. Weyl’s work provides a general test for this type of equidistribution and extends it to much richer arithmetic sequences.\n\n## Listen for frequencies instead of inspecting every arc\n\nWrap each number modulo one, so only its fractional part determines the position. An exponential sum adds arrows pointing at those positions, with an integer frequency controlling how many times the circle is traversed. The criterion requires the average arrow to tend to zero at every nonzero integer frequency. Cancellation of all those frequencies is equivalent to the correct long-run counts in every interval.\n\n## A deliberately failing classroom example\n\nTake the fractional parts of n divided by four. They cycle through one quarter, one half, three quarters and zero. At frequency one, four arrows cancel exactly, so a single frequency check looks excellent. At frequency four, every arrow points in the same direction, and their average is one rather than zero. The sequence is not equidistributed over the whole circle. This example explains why the theorem says every nonzero integer frequency.\n\n## What the polynomial theorem changes\n\nWeyl proves cancellation for polynomial sequences with an irrational nonconstant coefficient, then applies the criterion to obtain equidistribution. A constant irrational offset alone is insufficient. The original preserves earlier linear-case work by Bohl, Sierpiński and Weyl, as well as Hardy–Littlewood’s polynomial developments. The result is about asymptotic frequencies; it does not create independent samples or promise an optimal finite-sample discrepancy.\n\n## A transparent exercise for your agent\n\nBuild a small circle plot for the rational four-cycle and an irrational-step illustration. Record the step, number of marks and tested frequencies. For the four-cycle, check the cancellation exactly rather than trusting rounded complex arithmetic. For the irrational example, label finite observations as illustrations. A reusable contribution should distinguish a visible pattern, a tested finite frequency set and the theorem’s unrestricted limiting statement.\n\n## Scientific scope\n\nA qualitative asymptotic theorem for each sequence or fixed polynomial under the stated irrationality conditions, not a uniform numerical discrepancy rate over all irrational coefficients or a finite-sample randomness guarantee. An irrational constant term alone is insufficient. The scalar linear case predates this paper and is credited jointly in the original to Bohl, Sierpiński and Weyl; Hardy–Littlewood's earlier and parallel polynomial work is explicitly retained. Later metric theorems and the geometric appendix lie outside this record.\n\n## Sources for this explanation\n\n- [Über die Gleichverteilung von Zahlen mod. Eins](https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0077/LOG_0036.pdf): Pinned original contribution and stated hypotheses or experimental limits.\n",
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      "title": "Weyl: testing equidistribution by exponential sums and applying it to polynomials"
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    "what_changed": "A deterministic sequence can spread around a circle as evenly as interval lengths predict. Weyl’s exponential-sum criterion tests this by asking whether each nonzero integer frequency cancels out on average. His 1916 paper applies the method to polynomial sequences with an irrational nonconstant coefficient and to multidimensional settings under the stated independence conditions. Even distribution is a long-run frequency statement; it does not make the sequence independent, random or perfectly balanced after a fixed number of terms.",
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        "title": "A pattern can look evenly scattered",
        "text": "Walk around a circle by the same irrational fraction of a turn each time and mark your positions. No random-number generator is involved. Nevertheless, over the long run an arc occupying a tenth of the circle receives a tenth of the marks. Weyl’s work provides a general test for this type of equidistribution and extends it to much richer arithmetic sequences."
      },
      {
        "title": "Listen for frequencies instead of inspecting every arc",
        "text": "Wrap each number modulo one, so only its fractional part determines the position. An exponential sum adds arrows pointing at those positions, with an integer frequency controlling how many times the circle is traversed. The criterion requires the average arrow to tend to zero at every nonzero integer frequency. Cancellation of all those frequencies is equivalent to the correct long-run counts in every interval."
      },
      {
        "title": "A deliberately failing classroom example",
        "text": "Take the fractional parts of n divided by four. They cycle through one quarter, one half, three quarters and zero. At frequency one, four arrows cancel exactly, so a single frequency check looks excellent. At frequency four, every arrow points in the same direction, and their average is one rather than zero. The sequence is not equidistributed over the whole circle. This example explains why the theorem says every nonzero integer frequency."
      },
      {
        "title": "What the polynomial theorem changes",
        "text": "Weyl proves cancellation for polynomial sequences with an irrational nonconstant coefficient, then applies the criterion to obtain equidistribution. A constant irrational offset alone is insufficient. The original preserves earlier linear-case work by Bohl, Sierpiński and Weyl, as well as Hardy–Littlewood’s polynomial developments. The result is about asymptotic frequencies; it does not create independent samples or promise an optimal finite-sample discrepancy."
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        "text": "Build a small circle plot for the rational four-cycle and an irrational-step illustration. Record the step, number of marks and tested frequencies. For the four-cycle, check the cancellation exactly rather than trusting rounded complex arithmetic. For the irrational example, label finite observations as illustrations. A reusable contribution should distinguish a visible pattern, a tested finite frequency set and the theorem’s unrestricted limiting statement."
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