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The answer to the question is negative. Körner in Hausdorff dimension of sums of sets with themselves and Schmeling-Shmerkin in On the dimension of iterated sumsets showed that for any increasing sequence $\alpha_1 < \alpha_2 < \ldots < 1$, there is a compact set $A$, s.t. for every $k$, we have $\dim_{H} k A = \alpha_k$, where $kA = A + A + \ldots + A$ --- Minkowski sum of the set with itself $k$ times, and $\dim_H$ is the Hausdorff dimension of the set. In particular, choosing any sequence such that $\dim_H kA < 1$ for all $k$, none of the sum sets $kA$ could contain an interval.

See also the recent paper Dimension growth for iterated sumsets for some conditions under which we have dimension growth $\dim_H (A + A) > \dim_H(A)$. Unfortunately, it seems that the bounds there still fall short from implying that for some $k$, $\dim_H k A = 1$, much less providing quantitative bound on $k$.

On the positive side, Marstand in "Some Fundamental Geometrical Properties of Plane Sets of Fractional Dimensions" showed that for any given pair of sets $A,B$, and almost every angle $\theta$, if we write $\lambda_1 = \cos \theta, \lambda_2 = \sin \theta$, then $$ \dim_H (A_{\lambda_1} + B_{\lambda_2}) = \min(\dim_H(A) + \dim_H(B),1),$$ where $A_{\lambda}$ is the dilatation $A_{\lambda} := \{ \lambda x : x \in A\}$. In fact a slightly stronger statement holds: if $\dim_H (A) = \alpha$, there is a sequence of parameters $\lambda_1, \lambda_2, \ldots, \lambda_k > 0$ for $k = \lceil 1/\alpha \rceil + 1$, such that for $B := A_{\lambda_1} + \ldots + A_{\lambda_k}$ not only $\dim_H(B)=1$, but indeed the 1-dimensional measure of this set is positive, and therefore $B+B$ contains an interval (a generic sequence of $\lambda_i$ will do the job). Unfortunately, this result does not say anything about any specific sequence of $\lambda_i$ (of which $\lambda_1 = \ldots = \lambda_k$ is the most interesting). A more modern exposition of this theorem can be found in Chapter 9 of "Geometry of sets and measures in Euclidean spaces" by Pertti Mattila.

The answer to the question is negative. Körner in Hausdorff dimension of sums of sets with themselves and Schmeling-Shmerkin in On the dimension of iterated sumsets showed that for any increasing sequence $\alpha_1 < \alpha_2 < \ldots < 1$, there is a compact set $A$, s.t. for every $k$, we have $\dim_{H} k A = \alpha_k$, where $kA = A + A + \ldots + A$ --- Minkowski sum of the set with itself $k$ times, and $\dim_H$ is the Hausdorff dimension of the set. In particular, choosing any sequence such that $\dim_H kA < 1$ for all $k$, none of the sum sets $kA$ could contain an interval.

See also the recent paper Dimension growth for iterated sumsets for some conditions under which we have dimension growth $\dim_H (A + A) > \dim_H(A)$. Unfortunately, it seems that the bounds there still fall short from implying that for some $k$, $\dim_H k A = 1$, much less providing quantitative bound on $k$.

On the positive side, Marstand in "Some Fundamental Geometrical Properties of Plane Sets of Fractional Dimensions" showed that for any given pair of sets $A,B$, and almost every angle $\theta$, if we write $\lambda_1 = \cos \theta, \lambda_2 = \sin \theta$, then $$ \dim_H (A_{\lambda_1} + B_{\lambda_2}) = \min(\dim_H(A \times B),1) \geq \min(\dim_H(A) + \dim_H(B), 1),$$ where $A_{\lambda}$ is the dilatation $A_{\lambda} := \{ \lambda x : x \in A\}$. In fact a slightly stronger statement holds: if $\dim_H (A) = \alpha$, there is a sequence of parameters $\lambda_1, \lambda_2, \ldots, \lambda_k > 0$ for $k = \lceil 1/\alpha \rceil + 1$, such that for $B := A_{\lambda_1} + \ldots + A_{\lambda_k}$ not only $\dim_H(B)=1$, but indeed the 1-dimensional measure of this set is positive, and therefore $B+B$ contains an interval (a generic sequence of $\lambda_i$ will do the job). Unfortunately, this result does not say anything about any specific sequence of $\lambda_i$ (of which $\lambda_1 = \ldots = \lambda_k$ is the most interesting). A more modern exposition of this theorem can be found in Chapter 9 of "Geometry of sets and measures in Euclidean spaces" by Pertti Mattila.

The answer to the question is negative. Körner in Hausdorff dimension of sums of sets with themselves and Schmeling-Shmerkin in On the dimension of iterated sumsets showed that for any increasing sequence $\alpha_1 < \alpha_2 < \ldots < 1$, there is a compact set $A$, s.t. for every $k$, we have $\dim_{H} k A = \alpha_k$, where $kA = A + A + \ldots + A$ --- Minkowski sum of the set with itself $k$ times, and $\dim_H$ is the Hausdorff dimension of the set. In particular, choosing any sequence such that $\dim_H kA < 1$ for all $k$, none of the sum sets $kA$ could contain an interval.

See also the recent paper Dimension growth for iterated sumsets for some conditions under which we have dimension growth $\dim_H (A + A) > \dim_H(A)$. Unfortunately, it seems that the bounds there still fall short from implying that for some $k$, $\dim_H k A = 1$, much less providing quantitative bound on $k$.

On the positive side, Marstand in "Some Fundamental Geometrical Properties of Plane Sets of Fractional Dimensions" showed that for any given pair of sets $A,B$, and almost every angle $\theta$, if we write $\lambda_1 = \cos \theta, \lambda_2 = \sin \theta$, then $$ \dim_H (A_{\lambda_1} + B_{\lambda_2}) = \min(\dim_H(A) + \dim_H(B),1),$$ where $A_{\lambda}$ is the dilatation $A_{\lambda} := \{ \lambda x : x \in A\}$. In fact a slightly stronger statement holds: if $\dim_H (A) = \alpha$, there is a sequence of parameters $\lambda_1, \lambda_2, \ldots, \lambda_k > 0$ for $k = \lceil 1/\alpha \rceil + 1$, such that not only $\dim_H(A_{\lambda_1} + \ldots + A_{\lambda_k})=1$, but indeed the 1-dimensional measure of this set is positive, and therefore it contains an interval (a generic sequence of $\lambda_i$ will do the job). Unfortunately, this result does not say anything about any specific sequence of $\lambda_i$ (of which $\lambda_1 = \ldots = \lambda_k$ is the most interesting). A more modern exposition of this theorem can be found in Chapter 9 of "Geometry of sets and measures in Euclidean spaces" by Pertti Mattila.

The answer to the question is negative. Körner in Hausdorff dimension of sums of sets with themselves and Schmeling-Shmerkin in On the dimension of iterated sumsets showed that for any increasing sequence $\alpha_1 < \alpha_2 < \ldots < 1$, there is a compact set $A$, s.t. for every $k$, we have $\dim_{H} k A = \alpha_k$, where $kA = A + A + \ldots + A$ --- Minkowski sum of the set with itself $k$ times, and $\dim_H$ is the Hausdorff dimension of the set. In particular, choosing any sequence such that $\dim_H kA < 1$ for all $k$, none of the sum sets $kA$ could contain an interval.

See also the recent paper Dimension growth for iterated sumsets for some conditions under which we have dimension growth $\dim_H (A + A) > \dim_H(A)$. Unfortunately, it seems that the bounds there still fall short from implying that for some $k$, $\dim_H k A = 1$, much less providing quantitative bound on $k$.

On the positive side, Marstand in "Some Fundamental Geometrical Properties of Plane Sets of Fractional Dimensions" showed that for any given pair of sets $A,B$, and almost every angle $\theta$, if we write $\lambda_1 = \cos \theta, \lambda_2 = \sin \theta$, then $$ \dim_H (A_{\lambda_1} + B_{\lambda_2}) = \min(\dim_H(A) + \dim_H(B),1),$$ where $A_{\lambda}$ is the dilatation $A_{\lambda} := \{ \lambda x : x \in A\}$. In fact a slightly stronger statement holds: if $\dim_H (A) = \alpha$, there is a sequence of parameters $\lambda_1, \lambda_2, \ldots, \lambda_k > 0$ for $k = \lceil 1/\alpha \rceil + 1$, such that for $B := A_{\lambda_1} + \ldots + A_{\lambda_k}$ not only $\dim_H(B)=1$, but indeed the 1-dimensional measure of this set is positive, and therefore $B+B$ contains an interval (a generic sequence of $\lambda_i$ will do the job). Unfortunately, this result does not say anything about any specific sequence of $\lambda_i$ (of which $\lambda_1 = \ldots = \lambda_k$ is the most interesting). A more modern exposition of this theorem can be found in Chapter 9 of "Geometry of sets and measures in Euclidean spaces" by Pertti Mattila.

The answer to the question is negative. Körner in Hausdorff dimension of sums of sets with themselves and Schmeling-Shmerkin in On the dimension of iterated sumsets showed that for any increasing sequence $\alpha_1 < \alpha_2 < \ldots < 1$, there is a compact set $A$, s.t. for every $k$, we have $\dim_{H} k A = \alpha_k$, where $kA = A + A + \ldots + A$ --- Minkowski sum of the set with itself $k$ times, and $\dim_H$ is the Hausdorff dimension of the set. In particular, choosing any sequence such that $\dim_H kA < 1$ for all $k$, none of the sum sets $kA$ could contain an interval.

See also the recent paper Dimension growth for iterated sumsets for some conditions under which we have dimension growth $\dim_H (A + A) > \dim_H(A)$. Unfortunately, it seems that the bounds there still fall short from implying that for some $k$, $\dim_H k A = 1$, much less providing quantitative bound on $k$.

On the positive side, Marstand in "Some Fundamental Geometrical Properties of Plane Sets of Fractional Dimensions" showed that for any given pair of sets $A,B$, and almost every angle $\theta$, if we write $\lambda_1 = \cos \theta, \lambda_2 = \sin \theta$, then $$ \dim_H (A_{\lambda_1} + B_{\lambda_2}) = \min(\dim_H(A) + \dim_H(B),1),$$ where $A_{\lambda}$ is the dilatation $A_{\lambda} := \{ \lambda x : x \in A\}$. In particular, if $\dim_H (A) = \alpha$, there is a sequence of parameters $\lambda_1, \lambda_2, \ldots, \lambda_k > 0$ for $k = 2 \lceil 1/\alpha \rceil$, such that $A_{\lambda_1} + \ldots + A_{\lambda_k}$ contain an interval (and indeed, a generic sequence will do the job). Unfortunately, this result does not say anything about any specific sequence of $\lambda_i$ (of which $\lambda_1 = \ldots = \lambda_k$ is the most interesting).

The answer to the question is negative. Körner in Hausdorff dimension of sums of sets with themselves and Schmeling-Shmerkin in On the dimension of iterated sumsets showed that for any increasing sequence $\alpha_1 < \alpha_2 < \ldots < 1$, there is a compact set $A$, s.t. for every $k$, we have $\dim_{H} k A = \alpha_k$, where $kA = A + A + \ldots + A$ --- Minkowski sum of the set with itself $k$ times, and $\dim_H$ is the Hausdorff dimension of the set. In particular, choosing any sequence such that $\dim_H kA < 1$ for all $k$, none of the sum sets $kA$ could contain an interval.

See also the recent paper Dimension growth for iterated sumsets for some conditions under which we have dimension growth $\dim_H (A + A) > \dim_H(A)$. Unfortunately, it seems that the bounds there still fall short from implying that for some $k$, $\dim_H k A = 1$, much less providing quantitative bound on $k$.

On the positive side, Marstand in "Some Fundamental Geometrical Properties of Plane Sets of Fractional Dimensions" showed that for any given pair of sets $A,B$, and almost every angle $\theta$, if we write $\lambda_1 = \cos \theta, \lambda_2 = \sin \theta$, then $$ \dim_H (A_{\lambda_1} + B_{\lambda_2}) = \min(\dim_H(A) + \dim_H(B),1),$$ where $A_{\lambda}$ is the dilatation $A_{\lambda} := \{ \lambda x : x \in A\}$. In fact a slightly stronger statement holds: if $\dim_H (A) = \alpha$, there is a sequence of parameters $\lambda_1, \lambda_2, \ldots, \lambda_k > 0$ for $k = \lceil 1/\alpha \rceil + 1$, such that not only $\dim_H(A_{\lambda_1} + \ldots + A_{\lambda_k})=1$, but indeed the 1-dimensional measure of this set is positive, and therefore it contains an interval (a generic sequence of $\lambda_i$ will do the job). Unfortunately, this result does not say anything about any specific sequence of $\lambda_i$ (of which $\lambda_1 = \ldots = \lambda_k$ is the most interesting). A more modern exposition of this theorem can be found in Chapter 9 of "Geometry of sets and measures in Euclidean spaces" by Pertti Mattila.