TY - GEN
T1 - Multiparameter Projection Theorems with Applications to Sums-Products and Finite Point Configurations in the Euclidean Setting
AU - Erdoǧan, Burk
AU - Hart, Derrick
AU - Iosevich, Alex
PY - 2013
Y1 - 2013
N2 - In this paper we study multi-parameter projection theorems for fractal sets. With the help of these estimates, we recover results about the size of A · A +...+ A · A, where A is a subset of the real line of a given Hausdorff dimension, A + A = {a + a': a,a' ∈ A} and A·A = {a· a': a,a' ∈ A}. We also use projection results and inductive arguments to show that if a Hausdorff dimension of a subset of ℝd is sufficiently large, then the (k+1/2)-dimensional Lebesgue measure of the set of k-simplexes determined by this set is positive. The sharpness of these results and connection with number theoretic estimates is also discussed.
AB - In this paper we study multi-parameter projection theorems for fractal sets. With the help of these estimates, we recover results about the size of A · A +...+ A · A, where A is a subset of the real line of a given Hausdorff dimension, A + A = {a + a': a,a' ∈ A} and A·A = {a· a': a,a' ∈ A}. We also use projection results and inductive arguments to show that if a Hausdorff dimension of a subset of ℝd is sufficiently large, then the (k+1/2)-dimensional Lebesgue measure of the set of k-simplexes determined by this set is positive. The sharpness of these results and connection with number theoretic estimates is also discussed.
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U2 - 10.1007/978-1-4614-4565-4_11
DO - 10.1007/978-1-4614-4565-4_11
M3 - Conference contribution
AN - SCOPUS:84883353213
SN - 9781461445647
T3 - Springer Proceedings in Mathematics and Statistics
SP - 93
EP - 103
BT - Recent Advances in Harmonic Analysis and Applications
PB - Springer
ER -