Erdos 1949 uniform distribution modulo 1 lacunary sequences

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theorem_1: States that for any lacunary sequence of positive numbers lambda_n and almost all theta, the discrepancy of theta lambda_n satisfies N D(N) = o(N^{1/2} log^{3/2} N (log log N)^{1/2} omega(N)) for every positive increasing omega tending to infinity.

theorem_2: States that for almost all theta >= 1 the sequence theta, theta^2, theta^3, and so on has discrepancy satisfying N D(N) = o(N^{1/2} log^{3/2} N (log log N)^{1/2} omega(N)) for every positive increasing omega tending to infinity.

theorem_3: States that if f(n, theta) on a <= theta <= b has first and second theta-derivatives growing by a factor at least 1 + delta in n, the first positive and the second nonnegative, then for almost all theta the sequence f(n, theta) satisfies the discrepancy bound (5).

theorem_4: States that under the hypotheses of Theorem 3 and for each constant K > 0, almost every theta has a constant C(theta) bounding the exponential sums of k f(n, theta) over n <= N by C(theta) N^{1/2} log^{1/2} N (log log N)^{1/2} omega(N) for all integers 1 <= k <= N^K.

theorem_5: States the paper's main theorem: if the subsequences of f(n, theta) along s residue classes satisfy Condition A and a series built from B_N^* and a sequence psi converges, then almost every theta gives N D(N) <= K_1 s^{1/2} N^{1/2} psi([(N-1)/s]+1) log N for all large N.

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P. Erdős, J. F. Koksma: On the uniform distribution modulo 1 of lacunary sequences, Nederl. Akad. Wetensch., Proc. 52 (1949), 264--273 = Indag. Math. 11 (1949), 79--88 (MR 11,14b; Zentralblatt 33,165).

For a sequence of positive numbers lambda_n satisfying the lacunarity condition lambda_{n+1} >= (1+delta) lambda_n, Theorem 1 shows that for almost all theta the discrepancy of (theta lambda_n) satisfies N D(N) = o(N^{1/2} log^{3/2} N (log log N)^{1/2} omega(N)) for any positive increasing omega tending to infinity, which the authors state is sharper than all known results; they cite Khintchine's Omega(N^{1/2} sqrt(log log N)) for lambda_n = 2^n to show that the exponent 1/2 on N cannot be improved. Theorem 2 gives the same bound for the sequence theta, theta^2, theta^3, ... for almost all theta >= 1, where Drewes had given the sharpest earlier estimate. Both are cases of Theorem 3, on sequences f(n, theta) whose first and second theta-derivatives grow geometrically in n, which is itself a case of the paper's main Theorem 5, a metric discrepancy theorem for sequences f(n, theta) under a monotonicity condition (Condition A) on differences of sums over r-tuples. Theorem 4 is the matching exponential-sum bound, a form of Lemma 2. The method bounds high moments of exponential sums and applies the Erdős--Turán inequality; the authors note that it meets great difficulties in the non-lacunary case, which they treat with another method in a following paper. Problem 992 does not cite this paper: its [ErKo49] is the authors' companion paper On the uniform distribution modulo 1 of sequences (f(n, theta)) (Nederl. Akad. Wetensch. Proc. 52 (1949), 851-854 = Indag. Math. 11 (1949), 299-302); this paper's lacunary bound is context for the problem's lacunary case.

Source: https://users.renyi.hu/~p_erdos/1949-10.pdf. No notice is printed; the hosting archive's site footer speaks for the site, not the paper (https://users.renyi.hu/~p_erdos/, prints "(C) 2005-2007 All rights reserved. All material on this site is for scientifics purposes only."); a Crossref query on 2026-10-02 found no record for the article, and the publisher's page was not consulted; the term is unstated.

Results. Labels are the print's; pages are in the Indag. Math. pagination, with the Proc. page in parentheses.

• Theorem 1 (p. 80, Proc. p. 265): for positive lambda_n with lambda_{n+1} >= (1+delta) lambda_n and any positive increasing omega tending to infinity, the discrepancy of (theta lambda_n) satisfies N D(N) = o(N^{1/2} log^{3/2} N (log log N)^{1/2} omega(N)) for almost all theta.
• Theorem 2 (p. 80, Proc. p. 265): for almost all theta >= 1 the sequence theta, theta^2, theta^3, ... satisfies the same bound (5).
• Theorem 3 (p. 81, Proc. p. 266): the bound (5) for almost all theta in [a, b] when f'(n+1, theta) >= (1+delta) f'(n, theta) > 0 and f''(n+1, theta) >= (1+delta) f''(n, theta) >= 0 on [a, b].
• Theorem 4 (p. 81, Proc. p. 266): under the hypotheses of Theorem 3, the exponential sums of k f(n, theta) over n <= N are at most C(theta) N^{1/2} log^{1/2} N (log log N)^{1/2} omega(N) for all 1 <= k <= N^K, for almost all theta.
• Theorem 5 (pp. 82--83, Proc. pp. 267--268): the main theorem, N D(N) <= K_1 s^{1/2} N^{1/2} psi([(N-1)/s]+1) log N for almost all theta and large N under Condition A and a convergent series (13); its Lemmas 1 and 2 are summarized on its page.

Lemma 3 (p. 86) is the Erdős--Turán inequality, quoted from Erdős and Turán (1948) and not proved here.

Read status. Claims checked for Theorems 1 to 5, read clause by clause on the print; the proofs in §§ 6--9 were followed for their structure only.

Bears on. #992: for a lacunary integer sequence, Theorem 1 gives a count discrepancy o(N^{1/2} (log N)^{3/2} (log log N)^{1/2} omega(N)) for almost all alpha, which is weaker than both bounds the problem asks about and answers neither; the paper does not mention the problem.

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