Hancl 1991 expression real numbers help infinite series

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Hančl, Jarosław, Expression of real numbers with the help of infinite series. Acta Arith. 59 (1991), no. 2, 97--104.

Hančl extends Erdős's notion of an irrationality sequence from integer to real sequences and studies when a prescribed real number can be written as a series of reciprocals 1/(a_n g_n) with g_n drawn from a fixed unbounded set S. Theorem 1 gives a sufficient condition (condition (1), comparing max_k (1/b_k - 1/b_{k+1})/a_n with the tail sum of 1/a_j) for every A in (0,K] to be representable as sum 1/(a_n g_n) when sum 1/a_n = K is finite, necessary as well when a_n is nondecreasing and max_k (1/b_k - 1/b_{k+1}) = 1 - 1/b_2, plus a variant when K is infinite. Theorem 2 handles sets S whose gaps are bounded by a positive integer D and sequences containing a subsequence c_n satisfying 2^{-2^{n-F(n)}} < K/c_n with sum 2^{-F(n)} finite, and Theorem 3 covers sets S with 1 > K >= 1 - b_{n-1}/b_n and liminf 1/(C_n K^n) > 0; the proofs are all by the same explicit greedy induction constructing the coefficients g_n one at a time. Definition 1 calls a sequence rational if some choice of positive integers b_n makes sum 1/(a_n b_n) rational, and irrational otherwise; Corollary 1 says any sequence satisfying Theorem 2's hypotheses with S = {1,2,3,...} is rational, and Corollary 2 says every sequence with limsup (log_2 log_2 c_n)/n < 1 is rational. This bears on problem [262]: since an irrationality sequence must not be rational, Corollary 2 forces limsup (log_2 log_2 a_n)/n >= 1, essentially pinning the slowest possible growth of an irrationality sequence at the doubly exponential rate 2^{2^n}; Hančl notes as an open remark that he does not know whether 2^{2^n/n} is rational.

Source: https://doi.org/10.4064/aa-59-2-97-104. The scan is image-only and its rendered first and last pages show no copyright or license line; the journal's record offers the PDF under the download link "Pobierz zgodnie z CC-BY", rendered "Free download under CC-BY license" on the English site, and names no version or URL for it (https://www.impan.pl/get/doi/10.4064/aa-59-2-97-104, read 2026-10-02): the Creative Commons Attribution license, with no version stated.

Bears on. #262

Results to transcribe.

• Theorem 1: If sum 1/a_n = K < infinity and S = {b_1=1 < b_2 < ...} tends to infinity, then condition (max_k (1/b_k - 1/b_{k+1}))/a_n <= sum_{j>n} 1/a_j for all n suffices (and, for nondecreasing a_n with max_k (1/b_k-1/b_{k+1}) = 1 - 1/b_2, is necessary) for every A with 0 < A <= K to be written as A = sum 1/(a_n g_n) with g_n in S.
• Theorem 2: Let S = {b_1 < b_2 < ...} be positive reals tending to infinity with a positive integer D bounding its gaps (printed as D > b_{n-1} - b_n for every n; the proof uses b_n - b_{n-1} < D). If a_n has a subsequence c_n with 2^{-2^{n-F(n)}} < K/c_n for a positive integer K and a positive real function F(n) < n with sum 2^{-F(n)} < infinity, then there is B > 0 such that every B_1 in (0,B] equals sum 1/(a_n g_n) for some g_n in S.
• Theorem 3: For S with 1 > K >= 1 - b_{n-1}/b_n for n >= n_0 and a subsequence C_n with liminf 1/(C_n K^n) > 0, there is B > 0 such that every B_1 in (0,B] is representable as sum 1/(a_n g_n), g_n in S.
• Corollary 1: A sequence satisfying the hypotheses of Theorem 2 with S = {1,2,3,...} is a rational sequence, i.e. some choice of positive integers b_n makes sum 1/(a_n b_n) rational.
• Corollary 2: Every sequence of positive reals with limsup (log_2 log_2 c_n)/n < 1 is a rational sequence; equivalently an irrationality sequence must satisfy limsup (log_2 log_2 a_n)/n >= 1.