<b>Digitally Restricted Integers and Related Problems</b>

<p dir="ltr">In this thesis we consider several problems involving <i>W</i>, the set of integers whose base-<i>b</i> representations only contain digits in some set <i>D</i>⊊{0,1,...,b-1} with |<i>D</i>|>1.</p><p dir="ltr">For some large <i>X</i>, let <i>S(X,y)</i> be the set of all integers less than <i>X </i>whose prime factors are all no larger than <i>y</i>. When <i>b=10</i> or <i>b</i> is large and <i>|D|=b-1</i>, we obtain two different results on the cardinality of the intersection of <i>W</i> with <i>S(X,y)</i>. When <i>X</i>^∂ ≥<i>y</i> ≥<i>exp</i>((loglog <i>X</i>)⁷), we show that the cardinality of the intersection is asymptotically equal to <i>X</i>^<em>(</em>^log|^<em>D</em>^|)/(log^<em>b)</em>^-1|<i>S(X,y)|</i> When on the other hand, exp((log <i>X</i>)^1-^ε)≥<i>y</i>≥ (log <i>X</i>)^<em>c</em> for some constant <i>c</i>, we obtain a similar asymptotic formula, but modified by the insertion of a positive constant, reflecting the product of local densities associated with this counting problem.</p><p dir="ltr">Using similar methods, we also find an asymptotic of similar shape for the number of values any polynomial <i>P</i>∈<b>Z</b>[<i>x</i>] takes which lie in <i>W</i>, provided |<i>D</i>|=<i>b</i>-1 and <i>b</i> is sufficiently large in terms of <i>P</i>.</p><p dir="ltr">In addition to these results, we use related but slightly different methods to show that almost all even integers in <i>W</i> are the sum of two primes.</p><p dir="ltr">Unrelatedly, we also computationally verify the conjecture regarding <i>g(k)</i> in Waring's problem, sometimes known as ``Waring's conjecture'', for all values of <i>k</i>≤ 85,899,345,920. This improves on work of Kubina and Wunderlich from 1990 obtaining the corresponding result for <i>k</i>≤471,600,000.</p>