Lagarias 2009 ternary expansions powers 2

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theorem_1_1: Lagarias's uniform bound for the truncated real doubling system: for each real lambda > 0, at most 25 X^0.9725 of the integers floor(lambda 2^n) with 1 <= n <= X have a ternary expansion omitting the digit 2, once X is large enough in terms of lambda.

theorem_1_2: Lagarias's construction showing the truncated count is not always bounded: an infinite sequence of exponents n_k, with n_1 = 2 and each n_k between two exponentials of n_{k-1}, for which uncountably many real lambda > 0 have every floor(lambda 2^{n_k}) omitting the digit 2 in base three.

theorem_1_3: Lagarias's dimension theorem for the truncated real doubling system: the set of real lambda > 0 with infinitely many floor(lambda 2^n) omitting the digit 2 in base three has Hausdorff dimension log_3 2, about 0.63092, and positive log_3 2-dimensional Hausdorff measure.

theorem_1_4: Lagarias's 3-adic counting bound: for each nonzero 3-adic integer lambda and each X >= 2, at most 2 X^alpha_0 exponents n <= X give a 3-adic expansion of lambda 2^n omitting the digit 2, where alpha_0 = log_3 2, extending Narkiewicz's bound for lambda = 1.

theorem_1_5: Lagarias's dimension bounds for the 3-adic approximations to the exceptional set: the lambda with at least one lambda 2^n omitting the digit 2 form a set of dimension log_3 2, those with two such n a set of dimension between (1/2) log_3 2 and 1/2, and those with three such n a set of dimension at least (1/6) log_3 2.

theorem_1_6: Lagarias's upper bound for intersections of two multiplicative translates of the 3-adic Cantor set: for a positive integer M that is not a power of 3, the 3-adic integers lambda with both lambda and M lambda omitting the digit 2 form a set of Hausdorff dimension at most 1/2.

theorem_1_7: Lagarias's sufficient condition for positive dimension: if some positive integer N omitting the digit 2 has every N M_i omitting the digit 2, then the set of 3-adic lambda with every M_i lambda omitting the digit 2 has Hausdorff dimension at least log_3 2 divided by the ceiling of log_3(N M_k).

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Lagarias, Jeffrey C., Ternary expansions of powers of 2. J. Lond. Math. Soc. (2) 79 (2009), no. 3, 562--588. doi:10.1112/jlms/jdn080.

Source: https://arxiv.org/abs/math/0512006. The copy read for this card is the arXiv version arXiv:math/0512006v4, dated July 11, 2008, 28 pages; theorem numbers and pages below follow it, not the journal print. The arXiv record names arXiv's non-exclusive distribution license (arXiv:math/0512006), every other right reserved.

The paper generalizes Erdős's question of when the ternary expansion of 2^n omits the digit 2, which Erdős conjectured does not happen for n >= 9 (p. 1), by inserting a parameter lambda into two doubling dynamical systems: the truncated real system x_n(lambda) = floor(lambda 2^n) for real lambda > 0 and the 3-adic system y_n(lambda) = lambda 2^n for lambda in Z_3, with lambda = 1 recovering the powers of 2. The paper calls the "Conjecture of Erdős" the weaker assertion that only finitely many n work (p. 1), which is Problem 406.

Theorem 1.1 gives a uniform bound: for each lambda > 0 the count of 1 <= n <= X with (floor(lambda 2^n))3 omitting the digit 2 is at most 25 X^{0.9725} for all X >= n_0(lambda). Theorem 1.2 shows the count is not always bounded, constructing an infinite sequence of exponents n_k, each between two exponentials of n{k-1}, along which uncountably many lambda > 0 have every floor(lambda 2^{n_k}) omitting the digit 2. Theorem 1.3 computes the Hausdorff dimension of the truncated real exceptional set as log_3 2 ≈ 0.63092. On the 3-adic side Theorem 1.4 bounds the count by 2 X^{alpha_0} with alpha_0 = log_3 2 for every nonzero lambda and X >= 2, extending Narkiewicz's bound N_1(X) <= 1.62 X^{alpha_0} for lambda = 1 (p. 1). Theorem 1.5 gives dimension alpha_0 for the set E^{(1)}(Z_3) of lambda with at least one lambda 2^n omitting the digit 2, bounds the dimension of E^{(2)}(Z_3) between (1/2) log_3 2 and 1/2, and that of E^{(3)}(Z_3) between (1/6) log_3 2 and dim_H E^{(2)}(Z_3); these sets contain the 3-adic exceptional set E(Z_3). Theorems 1.6 and 1.7 bound the Hausdorff dimension of intersections of multiplicative translates of the 3-adic Cantor set, the first giving the upper bound 1/2 used in Theorem 1.5. The paper also poses Conjecture A (p. 4) and Conjecture B (p. 5), that the untruncated real and the 3-adic exceptional sets have Hausdorff dimension zero, and Conjecture E (p. 7), that every finite digit pattern occurs in the base-q expansion of p^n for all large n when p and q are multiplicatively independent.

Read status. Claims checked for Theorems 1.1--1.7 (pp. 2--6), read clause by clause on the page images of the arXiv v4 edition; the proofs of Theorems 1.4, 1.5 and 1.7 (pp. 20--21, 24--25) were followed, the others read for structure only. Nothing here is independently reviewed. Printed discrepancies are recorded on the result pages: Theorem 1.4 counts n <= X while its proof counts 1 <= n <= X, display (1.16) prints a union for an intersection, and Theorem 1.7 prints its hypothesis on N with a union sign and a mismatched index.

Bears on. #406: the paper states (pp. 3, 5) that the problem's assertion is equivalent to 1 not lying in the untruncated real exceptional set E(R_+), and to 1 not lying in the 3-adic exceptional set E(Z_3). With lambda = 1, Theorem 1.1 bounds the number of exponents 1 <= n <= X with 2^n omitting the digit 2 by 25 X^{0.9725} for all large X, and Theorem 1.4 bounds the number of such n <= X by 2 X^{log_3 2} for every X >= 2. Theorem 1.3 and Theorem 1.5 measure the exceptional sets without deciding whether 1 belongs to them. None of the results decides the problem.

Results.

• Theorem 1.1 (p. 2): For each lambda > 0, #{n : 1 <= n <= X and (floor(lambda 2^n))_3 omits the digit 2} <= 25 X^{0.9725} for all sufficiently large X >= n_0(lambda).
• Theorem 1.2 (p. 3): There is an infinite sequence S = {n_k} with n_1 = 2 and 2^{(n_{k-1}+2k-7)/14} <= n_k <= 2^{27(n_{k-1}+2k+6)} such that the set of lambda > 0 with every floor(lambda 2^n), n in S, omitting the digit 2 is uncountable.
• Theorem 1.3 (p. 3): The set E_T(R_+) of lambda > 0 for which infinitely many floor(lambda 2^n) omit the digit 2 in base 3 has Hausdorff dimension log_3 2 = log 2 / log 3 ≈ 0.63092, with nonzero log_3 2-dimensional Hausdorff measure.
• Theorem 1.4 (p. 4): For each nonzero 3-adic integer lambda and X >= 2, #{n <= X : the 3-adic expansion of lambda 2^n omits the digit 2} <= 2 X^{alpha_0}, alpha_0 = log_3 2.
• Theorem 1.5 (pp. 4--5): dim_H E^{(1)}(Z_3) = alpha_0 ≈ 0.63092; (1/2) log_3 2 <= dim_H E^{(2)}(Z_3) <= 1/2; and (1/6) log_3 2 <= dim_H E^{(3)}(Z_3) <= dim_H E^{(2)}(Z_3), where E^{(k)}(Z_3) is the set of lambda with at least k values of lambda 2^n omitting the digit 2.
• Theorem 1.6 (p. 6): For a positive integer M that is not a power of 3, the set C(1, M) of lambda in Z_3 with lambda and M lambda both omitting the digit 2 has Hausdorff dimension at most 1/2.
• Theorem 1.7 (p. 6): For integers 1 <= M_1 < ... < M_k, if a positive integer N omitting the digit 2 has every N M_i omitting the digit 2, then dim_H C(M_1, ..., M_k) >= log_3 2 / ceil(log_3(N M_k)).

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